You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the square ABCD shown in the figure, G is the midpoint of side CD. Line AG is extended to intersect the diagonal BD at point F and the extended line of side BC at point E. If FG equals 2, what is the length of AE?
(A) 6
(B) 8
(C) 10
(D) 12

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, lines a, b, c, and d are shown. It is known that c is perpendicular to a and b, and lines b, c, and d intersect at one point. If angle 1 is 48°, what is the measure of angle 2?
Choices:
(A) 42°
(B) 48°
(C) 40°
(D) 45°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, AB is parallel to CD, ∠A is 50°, ∠C is 20°, what is the measure of ∠E?
Choices:
(A) 20°
(B) 30°
(C) 50°
(D) 70°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, it is known that a is parallel to b. Xiao Ming places the right angle vertex of the triangle board on line b. If ∠1 is 35°, what is the degree measure of ∠2?
Choices:
(A) 65°
(B) 120°
(C) 125°
(D) 145°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x from the given choices:
(A) 3
(B) 2 \sqrt { 3 }
(C) 6
(D) 4 \sqrt { 3 }

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, if the regular pentagon ABCDE is inscribed in circle O, what is the measure of angle DAE?
Choices:
(A) 36°
(B) 26°
(C) 30°
(D) 45°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, if equilateral triangle ABC with side length 5 units is translated 4 units to the right along side BC to form triangle A'B'C', what is the length of segment B'C?
Choices:
(A) 1
(B) 2
(C) 4
(D) 5

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x and round the side measurement to the nearest tenth.
Options:
(A) 5.4
(B) 7.7
(C) 8.3
(D) 11.9

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, point P outside circle O is connected to the two tangents PA and PB on circle O, with tangents at points A and B respectively. Point O intersects circle O at point C, and point D is a moving point on triangle ABC that does not coincide with points A or C. If ∠APB = 80°, what is the degree measure of ∠ADC?
Choices:
(A) 15°
(B) 20°
(C) 25°
(D) 30°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Calculate the area of the parallelogram and round to the nearest tenth if needed.
Choices:
(A) 41.0
(B) 57.9
(C) 70.9
(D) 81.9

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x.
Options:
(A) 5
(B) 5 \sqrt { 2 }
(C) 10
(D) 10 \sqrt { 2 }

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the quadrilateral ABCD, where AB=3, BC=5, ∠A=130°, ∠D=100°, and AD=CD. If points E and F are the midpoints of sides AD and CD respectively, what is the length of EF?
Choices:
(A) √{2}
(B) √{3}
(C) 2
(D) √{5}

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given figure, line l1 is parallel to line l2, ∠MAB = 125°, and ∠NBA = 85°. What is the degree measure of ∠1 + ∠2?
Choices:
(A) 30°
(B) 35°
(C) 36°
(D) 40°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, AB is the diameter of circle O, CD is a tangent to circle O with C as the point of tangency. A perpendicular line is drawn from A to CD with D as the foot of the perpendicular. If the radius of circle O is 10, CD is 8, what is the length of AD?
Choices:
(A) 12
(B) 18
(C) 14
(D) 16

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, with circle O, diameter AB is perpendicular to chord CD with foot E, angle A is 15°, and radius is 2. What is the length of chord CD?
Choices:
(A) 2
(B) 1
(C) √{2}
(D) 4

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, points A, B, C, D, E lie on circle O, with angle BAC measuring 10° and angle CED measuring 30°. What is the degree measure of angle BOD?
Choices:
(A) 40°
(B) 60°
(C) 80°
(D) 0°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, the perpendicular bisector of AB intersects AB at point D and intersects BC at point E, with AE being connected. If AB = 6 and the perimeter of triangle ACE is 13, what is the perimeter of triangle ABC?
Choices:
(A) 19
(B) 16
(C) 29
(D) 18

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, it is known that OA=OB=OC, BC is parallel to AO, and if ∠A=36°, then what is the measure of ∠B?
Choices:
(A) 54°
(B) 60°
(C) 72°
(D) 76°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, when a rectangular piece of paper is folded along EF, points D and C land at positions D' and C' respectively. If ∠EFB = 60°, what is the measure of ∠AED'?
Choices:
(A) 50°
(B) 55°
(C) 60°
(D) 65°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the rectangle ABCD, where AB=2 and BC=3, circles are drawn with centers at A and D with radii of 1. E and F are moving points on circles A and D respectively, while P is a moving point on BC. What is the minimum value of PE+PF?
Choices:
(A) 2
(B) 3
(C) 4
(D) 5

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, where AB is the diameter of circle O, angle B is 60°, and angle C is 70°, what is the degree measure of angle BOD?
Choices:
(A) 90°
(B) 100°
(C) 110°
(D) 120°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, AB is the diameter of circle O, AC is a tangent to circle O, and OC intersects circle O at point D. If ∠C = 42°, what is the degree measure of ∠ABD?
Choices:
(A) 48°
(B) 28°
(C) 34°
(D) 24°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, if the lengths of the diagonals AC and BD of rhombus ABCD are 6cm and 8cm respectively, what is the perimeter of this rhombus?
Choices:
(A) 5cm
(B) 10cm
(C) 14cm
(D) 20cm

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, point O is the incenter of triangle ABC. If ∠A = 70°, what is the measure of ∠BOC?
Choices:
(A) 120°
(B) 125°
(C) 130°
(D) 135°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, point P is outside circle O. PA and PB are tangents to circle O at points A and B respectively. AB and OP are connected, and OP intersects circle O at point D. If ∠APB = 56°, what is the measure of ∠ABD?
(A) 28°
(B) 30°
(C) 31°
(D) 34°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, if the vertex of the triangle board with a 60° angle is placed at the center of circle O, with the two sides intersecting circle O at points A and B, and point C on circle O, what is the degree measure of angle ACB?
Choices:
(A) 90°
(B) 60°
(C) 45°
(D) 30°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, to measure the height of a building CD, the angle of elevation to the top of the building C is 30° from point A on the ground. Moving 12 meters along the direction of AD to point B, the angle of elevation to the top of the building C is 60° from point B. What is the height of the building CD?
Choices:
(A) 6√{3} meters
(B) 6√{2} meters
(C) 3√{3} meters
(D) 4√{5} meters

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

During a weekend visit to Goose Ridge Park, Xiaojia observed the angles of elevation to the base and top of the Pisces Tower from point A as 12° and 13° respectively. The horizontal distance between points B and E was measured at 450 meters, with a slope ratio of 8:15 for the slope BC. Given that the horizontal distance AE was 1200 meters and the slope ratio of BC was 8:15, what is the height of the Pisces Tower CD in meters? (Reference data: tan 12° = 0.2, tan 13° = 0.23)
Choices:
(A) 34
(B) 35
(C) 36
(D) 37

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, points D and E are located on sides AB and AC respectively. Lines CD and BE intersect at point O, with DE parallel to BC. Given that OD = 1, OC = 3, and AD = 2, what is the length of side AB?
(A) 4
(B) 6
(C) 8
(D) 9

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, AB is the diameter of circle O. Points C and D are on the circle, with ∠CBA measuring 70°. What is the degree measure of angle D?
Choices:
(A) 10°
(B) 20°
(C) 70°
(D) 90°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, AB and BC are tangent to circle O at points A and B, and EF is tangent to circle O at point D. If BA = 20, what is the perimeter of triangle BEF?
Choices:
(A) 20
(B) 40
(C) 30
(D) 60

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the quadrilateral ABCD shown in the figure, with diagonal AC perpendicular to BD and with O as the foot of the perpendicular, points E, F, G, H are the midpoints of sides AB, BC, CD, AD respectively. If AC = 10 and BD = 6, what is the area of quadrilateral EFGH?
Choices:
(A) 15
(B) 20
(C) 30
(D) 60

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, AB is the diameter of circle O. Points C and D are on the circle on the same side of AB, and ∠ACD = 130°. What is the measure of ∠BAD?
Choices:
(A) 50°
(B) 40°
(C) 35°
(D) 25°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x.
Options:
(A) √6
(B) 6
(C) 6√3
(D) 12

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

If line m is parallel to line l and the measure of angle 1 is 73 degrees, which statement cannot be true based on the given conditions?
Choices:
(A) \angle 1 \cong \angle 4
(B) \angle 3 \cong \angle 1
(C) m \angle 2 + m \angle 3 = 180
(D) m \angle 4 > 73

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, AD is parallel to BE and CF, and lines l1 and l2 intersect these three parallel lines at points A, B, C and points D, E, F respectively. If AB = 2, AC = 6, and DE = 1.5, what is the length of DF?
Choices:
(A) 7.5
(B) 6
(C) 4.5
(D) 3

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, ∠MON = 60°. ① Using point O as the center, draw arcs with a radius of 2cm intersecting OM and ON at points A and C; ② Using A and C as centers, draw arcs with a radius of 2cm intersecting at point B; ③ Connect AB and BC, then the area of quadrilateral OABC is
Choices:
(A) 4√{3}
(B) 2√{3}
(C) 4
(D) 2

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, the radius of circle O is 5, chord AB = 8, point C is the midpoint of AB, and OC is connected. What is the length of OC?
Choices:
(A) 1
(B) 2
(C) 3
(D) 4

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, in circle O, OD is perpendicular to BC, and ∠BOD = 60°. What is the degree measure of ∠CAD?
Choices:
(A) 15°
(B) 20°
(C) 25°
(D) 30°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, where AB = AC, angle BAC is 100°, AD is the median on side BC, CE bisects angle BCA and intersects AB at point E. If AD and CE intersect at point F, what is the measure of angle CFA?
Choices:
(A) 100°
(B) 105°
(C) 110°
(D) 120°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, if one diagonal of a rhombus is 7 units long and the area is 28 square units, what is the length of the other diagonal of the rhombus?
Choices:
(A) 8
(B) 10
(C) 12
(D) 14

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure shown, PA and PB are tangents to circle O at points A and B, respectively. Point C lies on arc ACB, and angle C is 64°. What is the measure of angle P?
Choices:
(A) 26°
(B) 62°
(C) 65°
(D) 52°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the parallelogram ABCD shown in the figure, if ∠A = 150°, what is the measure of angle C?
Choices:
(A) 30°
(B) 75°
(C) 100°
(D) 150°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In right triangle ABC, with angle BAC measuring 90 degrees, AC equals 3 and AB equals 4. What is the cosine value of angle DAC?
Choices:
(A) 3/4
(B) 4/3
(C) 3/5
(D) 4/5

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given the figure where \triangle ABC is isosceles, \triangle DCE is equilateral, and \triangle FCG is isosceles, what is the measure of angle 3 at vertex C?
(A) 22
(B) 26
(C) 27
(D) 28

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, points A, B, and C are on circle O, with angle A measuring 35°. What is the measure of angle BOC?
Choices:
(A) 70°
(B) 45°
(C) 40°
(D) 35°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the rhombus ABCD shown in the figure, with ∠BAD = 110°, the perpendicular bisector of AB intersects the diagonal AC at point F, E is the foot, and DF is connected. What is the measure of ∠CDF?
Choices:
(A) 55°
(B) 45°
(C) 25°
(D) 15°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the area of the triangle shown below, rounding your answer to the nearest tenth if needed.
Choices:
(A) 110.5
(B) 144.2
(C) 164.5
(D) 171.9

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In right triangle ABC, where angle ACB is 90 degrees, three squares are constructed outside triangle ABC using its sides as edges. Let S1, S2, and S3 represent the areas of these three squares. If S1 = 3 and S2 = 11, what is the value of S3?
Choices:
(A) 5
(B) 8
(C) 14
(D) 16

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, AB is a diameter, point D lies on the extension of AB, and DC is tangent to circle O at C. If angle A is 25°, what is the measure of angle D?
Choices:
(A) 25°
(B) 40°
(C) 50°
(D) 65°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, in quadrilateral ABCD, AB = 4, BC = 7, and the angle bisector of ∠ABC intersects AD at point E. What is the length of DE?
Choices:
(A) 5
(B) 4
(C) 3
(D) 2

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, it is known that AB + AC = 16. Point O is the intersection of the bisectors of ∠ABC and ∠ACB, and OD is perpendicular to BC at D. If OD = 4, what is the area of quadrilateral ABOC?
Choices:
(A) 36
(B) 32
(C) 30
(D) 64

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, line a is parallel to line b, with vertex C of right triangle ABC on lines a and b, and ∠ACB = 90°. If ∠1 = 60°, what is the degree measure of ∠2?
Choices:
(A) 60°
(B) 30°
(C) 40°
(D) 20°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In a kite C D F G, what is the measure of angle D?
Choices:
(A) 18
(B) 49
(C) 67
(D) 122

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, PA and PB are tangents to circle O, with points of tangency A and B respectively. If ∠APB = 50° and point C is on circle O, what is the measure of angle ACB?
Choices:
(A) 50°
(B) 55°
(C) 60°
(D) 65°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, the diameter CD of circle O passes through the midpoint G of chord EF, ∠EOD=50°, ∠DCF is equal to ()
Choices:
(A) 80°
(B) 50°
(C) 40°
(D) 25°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In order to measure the height of the oil surface in the barrel, a thin wooden stick is inserted into the barrel from a small hole at the edge. The length of the inserted part of the stick is 100cm, the part of the stick with oil is 60cm, and the height of the barrel is 80cm. What is the height of the oil surface inside the barrel?
Choices:
(A) 32cm
(B) 30cm
(C) 50cm
(D) 48cm

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the quadrilateral ABCD shown, where AB is parallel to CD, AB is 5, DC is 11, the sum of AD and BC is 12. Points E, F, G are the midpoints of BD, AC, and DC respectively. What is the perimeter of triangle EFG?
Choices:
(A) 8
(B) 9
(C) 10
(D) 12

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, points D and E lie on the extensions of sides BA and CA of triangle ABC, respectively, with DE parallel to BC. If EC is 3 times EA and the perimeter of triangle AED is 3, what is the perimeter of triangle ABC?
Choices:
(A) 3
(B) 6
(C) 9
(D) 12

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, point F lies on the diagonal BD of parallelogram ABCD. If BF:FD=1:3, what is the ratio of BE to EC?
Choices:
(A) 1/2
(B) 1/3
(C) 2/3
(D) 1/4

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, the diagonals AC and BD of quadrilateral ABCD intersect at O. Line EF passes through point O and intersects AD and BC at E and F respectively. If the area of ABCD is 20 cm, what is the area of the shaded region in the diagram?
Choices:
(A) 12
(B) 10
(C) 8
(D) 5

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, point C lies on the semicircle with diameter AB, where BE and AD are the angle bisectors of ∠ABC and ∠CAB respectively. If AB = 6, what is the length of DE?
Choices:
(A) 3
(B) 3√{2}
(C) 3√{3}
(D) 5

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, AB is the diameter of circle O, and C, D are two points on the circle. If ∠AOC = 50°, what is the measure of ∠D?
Choices:
(A) 25°
(B) 30°
(C) 40°
(D) 50°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, AB is the diameter of circle O, point C lies on the extension of AB, CD is tangent to circle O at point D. If ∠A = 25°, what is the degree measure of ∠C?
Choices:
(A) 40°
(B) 50°
(C) 65°
(D) 25°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, AB is the diameter of circle O, point D lies on the extension of AB, and DC is tangent to circle O at point C. If ∠A = 35°, what is the measure of ∠D?
Choices:
(A) 50°
(B) 40°
(C) 30°
(D) 20°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, angle BAC is 30°, AB = AC = 12. Point P is a moving point on side AB, and a parallelogram PAQC is constructed with sides PA and PC. What is the minimum length of diagonal PQ?
Choices:
(A) 6
(B) 12
(C) 4√{3}
(D) 6√{3}

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, the right angle vertex of the triangle board lies on one side of the rectangular paper. If ∠2 = 50°, then ∠1 = ?
Choices:
(A) 35°
(B) 40°
(C) 45°
(D) 50°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, AD is the angle bisector of triangle ABC, DF is perpendicular to AB with F as the foot of the perpendicular, DE is equal to DG, and the areas of triangles ADG and AED are 29 and 16 respectively. What is the area of triangle EDF?
Choices:
(A) 13
(B) 6.5
(C) 11
(D) 5.5

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, quadrilateral ABCD is inscribed in circle O, AD is the diameter, ∠ABC = 120°, CD = 3. What is the length of chord AC?
Choices:
(A) 3√{3}
(B) 2√{3}
(C) √{3}
(D) 4

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, AB and AC are chords of circle O, with OE perpendicular to AB and OF perpendicular to AC, and E and F as the respective feet of the perpendiculars. If EF = 3.5, what is the length of BC?
Choices:
(A) 3.5
(B) 7
(C) 10.5
(D) 9

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, line AB is parallel to line CD, and line EF intersects line AB and CD at points O and G, respectively. Line OP bisects angle EOB, and if angle EOP is 35°, what is the measure of angle DGF?
Choices:
(A) 90°
(B) 100°
(C) 110°
(D) 120°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, quadrilateral ABCD is inscribed in circle O. If ∠B = 70°, what is the measure of ∠D?
Choices:
(A) 70°
(B) 110°
(C) 35°
(D) Insufficient information to calculate

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In rectangle JKLM, if the measure of angle KJL is 2x + 4 and the measure of angle JLK is 7x + 5, what is the value of x?
Choices:
(A) 9
(B) 22
(C) 45
(D) 68

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, there is oil in a barrel with a height of 1m. A wooden stick, 1.2m long, is inserted diagonally into the barrel from the small opening in the lid, with one end reaching the bottom and the other end just reaching the opening. After removing the stick, the length of the oil-soaked part on the stick is measured to be 0.36m. What is the height of the oil in the barrel?
Choices:
(A) 0.28m
(B) 0.385m
(C) 0.4m
(D) 0.3m

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, line a is parallel to line b and intersected by line c. If angle 1 is 110°, what is the degree measure of angle 2?
Choices:
(A) 108°
(B) 72°
(C) 70°
(D) 60°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In rhombus ABCD, where AC=2 and BD=5, point P is any point on diagonal AC. Line PE is drawn parallel to AD and PF is drawn parallel to AB, intersecting AB and AD at points E and F respectively. What is the area of the shaded region in the figure?
(A) 2
(B) 5/2
(C) 5
(D) 10

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x from the given choices.
(A) 8 \sqrt { 6 }
(B) 20
(C) 28
(D) 4 \sqrt { 74 }

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, line ED is perpendicular and bisects segment BC, intersecting BC and AB at points D and E respectively. If BD = 3 and the perimeter of triangle AEC is 20, what is the perimeter of triangle ABC?
(A) 23
(B) 26
(C) 28
(D) 30

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, point D is on side AB. If angle ACD is equal to angle B, AD is 1, AC is 2, and the area of triangle ADC is 1, what is the area of triangle BCD?
Choices:
(A) 1
(B) 2
(C) 3
(D) 4

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, if l_{1}∥l_{2}∥l_{3}, AB=3, AC=15, DE=2, what is the length of EF?
Choices:
(A) 8
(B) 10
(C) 4
(D) 6

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the circle with radius 1, extend the diameter AB to point C such that BC = OA. CD is a tangent to the circle at point D. What is the length of AD?
Choices:
(A) 1
(B) √{3}
(C) 2
(D) 2√{3}

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, line segment AB is the diameter of circle O, chord CD is perpendicular to AB, and ∠CAB = 20°. What is the measure of ∠AOD?
Choices:
(A) 120°
(B) 140°
(C) 150°
(D) 160°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, a rope is represented by line segment AB. The rope is cut at point C, resulting in two segments with the longer one measuring 60cm. If AC is 3/4 of CB, what was the original length of the rope?
Choices:
(A) 140cm
(B) 45cm
(C) 105cm
(D) 90cm

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the pattern shown, which is a symmetrical figure with line l as its axis of symmetry, if the radius of the largest circle is 2, what is the area of the shaded region?
Choices:
(A) π
(B) 2π
(C) 3π
(D) 4π

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of h in the triangle.
Choices:
(A) 3 \sqrt 3
(B) 6
(C) 6 \sqrt 2
(D) 6 \sqrt 3

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, in circle O, OA is perpendicular to BC at E, and ∠AOB = 50°. What is the measure of ∠ADC?
Choices:
(A) 25°
(B) 30°
(C) 40°
(D) 50°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the measure of angle 2.
Options:
(A) 29
(B) 52
(C) 57
(D) 123

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, triangle ABC is inscribed in circle O, AB = AC, P is a point on side AB, and ∠BAC = 30°. What is the measure of ∠APB?
(A) 105°
(B) 110°
(C) 115°
(D) 120°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, BC is the diameter of circle O, and AB is a chord of circle O. If ∠AOC = 60°, what is the degree measure of ∠OAB?
Choices:
(A) 20°
(B) 25°
(C) 30°
(D) 35°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, angle B is 70°. If point B is folded along the dashed line EF to point D on AC, what is the sum of angles 1 and 2?
Choices:
(A) 160°
(B) 150°
(C) 140°
(D) 110°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the quadrilateral ABCD shown in the figure, where AD is parallel to BC and AC and BD intersect at point O, EF is drawn parallel to AD passing through point O and intersecting AB at points E and F. If AE = 2, BE = 5, and OD = 3, what is the length of BD?
Choices:
(A) 6
(B) 15/2
(C) 21/2
(D) 10/3

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, in circle O, if ∠BOC = 80°, then what is the measure of ∠A?
Choices:
(A) 20°
(B) 30°
(C) 40°
(D) 50°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, line a is parallel to line b, point B lies on line a, and AB is perpendicular to BC. If angle 1 is 35°, what is the measure of angle 2?
Choices:
(A) 35°
(B) 50°
(C) 55°
(D) 65°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, AB is the diameter of a semicircle, point D is the midpoint of segment AC, and ∠ABC = 50°. What is the measure of ∠DAB?
Choices:
(A) 65°
(B) 60°
(C) 55°
(D) 50°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x.
Options:
(A) 2√3
(B) 3√3
(C) 5√3
(D) 6√3

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, in quadrilateral ABCD, BF bisects angle ABC, intersecting AD at point F, and CE bisects angle BCD, intersecting AD at point E. If AB = 6 and BC = 10, what is the length of EF?
Choices:
(A) 1
(B) 2
(C) 3
(D) 4

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, a cone is formed by surrounding a circle with sector paper with a radius of 10 and a central angle of 144° (ignoring the seam). What is the radius of the base of this cone?
Choices:
(A) 3
(B) 4
(C) 5
(D) 6

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x.
Options:
(A) 3
(B) 4
(C) 6
(D) 12

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

What is the decimal equivalent, rounded to the nearest hundredth, of the ratio of tangent Q?
Choices:
(A) 0.47
(B) 0.53
(C) 0.88
(D) 1.88

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, BF bisects angle ABC, AF is perpendicular to BF at point F, D is the midpoint of AB, and line DF extended intersects AC at point E. If AB = 10 and BC = 16, what is the length of segment EF?
Choices:
(A) 2
(B) 3
(C) 4
(D) 5

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, there are points A and B on the outer edge of the protractor, with readings of 70° and 40° respectively. What is the degree measure of angle 1?
Choices:
(A) 15°
(B) 30°
(C) 40°
(D) 70°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, in quadrilateral ABCD with a perimeter of 18cm, where AB is not equal to AD, AC and BD intersect at point O, and OE is perpendicular to BD at E, what is the perimeter of triangle ABE?
(A) 8cm
(B) 9cm
(C) 10cm
(D) 12cm

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In trapezoid JKLM, the midpoints of the legs are points A and B. If AB measures 57 and KL measures 21, what is the length of JM?
Choices:
(A) 21
(B) 57
(C) 87
(D) 93

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Calculate the area of the shaded sector and round to the nearest tenth, if needed.
Choices:
(A) 15.3
(B) 52.6
(C) 82.4
(D) 284.0

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, AB is perpendicular to CD at D, DE is perpendicular to DF. If ∠BDE = 60°, what is the degree measure of ∠ADF?
Choices:
(A) 30°
(B) 45°
(C) 60°
(D) 120°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x.
Options:
(A) 5
(B) 10
(C) 20
(D) 100

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, quadrilateral ABCD is inscribed in circle O. If ∠A = 62°, what is the measure of ∠BCE?
Choices:
(A) 28°
(B) 31°
(C) 62°
(D) 118°

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, it is known that AB is the diameter of circle O, and ∠AOC is 130°. What is the measure of angle D?
Choices:
(A) 65°
(B) 25°
(C) 15°
(D) 35°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

If triangles JKL and PQR are similar and the area of triangle JKL is 30 square inches, what is the area of triangle PQR?
(A) 19.2
(B) 24.0
(C) 37.5
(D) 46.9

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, the diagonals of quadrilateral ABCD intersect at point O, with AB not equal to AD. A line OE is drawn perpendicular to BD passing through point O and intersecting BC at point E. If the perimeter of triangle CDE is 8cm, what is the perimeter of quadrilateral ABCD?
Choices:
(A) 8cm
(B) 12cm
(C) 16cm
(D) 24cm

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, there are two villages, A and B, on opposite sides of a river. It is measured that points A, B, and D are on a straight line, while points A, C, and E are on another straight line. BC is parallel to DE, DE measures 90 meters, BC measures 70 meters, and BD measures 20 meters. What is the distance between villages A and B?
Choices:
(A) 50 meters
(B) 60 meters
(C) 70 meters
(D) 80 meters

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, where AB = AC = 13 and the area of the triangle is 65, point D is any point on side BC. What is the sum of the distances from point D to sides AB and AC?
Choices:
(A) 5
(B) 6.5
(C) 9
(D) 10

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, if DE is parallel to BC, AD = 3, AE = 2, and BD = 5, what is the length of AC?
(A) 16/3
(B) 10/3
(C) 5/3
(D) 15/2

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x from the given choices:
(A) 2√2
(B) 3√2
(C) 5√2
(D) 6√2

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x.
Options:
(A) 24
(B) 32
(C) 40
(D) 8 \sqrt { 41 }

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

If pentagon ABCDE∽pentagon POGMN and the ratio of their corresponding altitudes is 3:2, what is the ratio of the areas of pentagon ABCDE to pentagon POGMN?
Choices:
(A) 2:3
(B) 3:2
(C) 6:4
(D) 9:4

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, where angle C is 90° and angle BAC is 70°, if triangle ABC is rotated clockwise 70° around point A and points B and C are rotated to B' and C' respectively, forming line BB', what is the measure of angle B'BC'?
Choices:
(A) 35°
(B) 40°
(C) 50°
(D) 55°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the measure of angle J L K.
Choices:
(A) 30
(B) 40
(C) 50
(D) 70

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, ABCD is a quadrilateral inscribed in circle O, where AD is the diameter and ∠CBE = 50°. What is the degree measure of the central angle ∠AOC in the diagram?
Choices:
(A) 30°
(B) 20°
(C) 50°
(D) 100°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Given that line m is parallel to line n, a right-angled triangle ABC with a 30° angle at point B is placed as shown in the diagram, where points A and B lie on lines m and n, respectively. If angle 1 is 20°, what is the degree measure of angle 2?
Choices:
(A) 20°
(B) 30°
(C) 45°
(D) 50°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the diagram, BD is the median of triangle ABC, with points E and F as the midpoints of BD and CE, respectively. If the area of triangle ABC is 12, what is the area of triangle AEF?
Choices:
(A) 2
(B) 3
(C) 4
(D) 6

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

If a lumberjack saws a log as shown in the figure, with a radius of OA=100mm and a chord length of AB=160mm, what is the maximum height of the sawn part of the log in mm?
Choices:
(A) 40
(B) 60
(C) 80
(D) 100

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, A and D are two points on the circle O, with BC as the diameter. If ∠D = 32°, what is the measure of ∠OAC?
(A) 64°
(B) 58°
(C) 68°
(D) 55°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x using the properties of a parallelogram.
Choices:
(A) 39
(B) 79
(C) 101
(D) 180

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the circle O, if AB = AC and angle BAC is 70°, what is the degree measure of angle AEC?
Choices:
(A) 65°
(B) 75°
(C) 50°
(D) 55°

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the figure, circle O is the circumcircle of triangle ABC, with angle AOB measuring 50°. What is the degree measure of angle C?
Choices:
(A) 25°
(B) 30°
(C) 35°
(D) 40°

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the value of x using the provided areas for the pair of similar figures.
Choices:
(A) 6.0
(B) 10.4
(C) 31.2
(D) 54.0

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, where angle C is 90 degrees, AC is 2, and BC is 1, what is the value of cos A?
Choices:
(A) \frac{1}{2}
(B) \frac{√{5}}{2}
(C) \frac{√{5}}{5}
(D) \frac{2√{5}}{5}

Answer: $\boxed{D}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In the given diagram, points A, B, and C lie on circle O, with angle AOB measuring 30°. What is the degree measure of angle ACB?
Choices:
(A) 10°
(B) 15°
(C) 40°
(D) 70°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In a right-angled triangle board with angles of 30° and 45° as shown in the diagram, if DE is parallel to BC, what is the degree measure of ∠ABE?
Choices:
(A) 10°
(B) 15°
(C) 20°
(D) 25°

Answer: $\boxed{B}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Calculate the area of the shaded region, assuming all polygons that appear to be regular are indeed regular. Round your answer to the nearest tenth.
Choices:
(A) 114.2
(B) 214.2
(C) 264.2
(D) 314.2

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Determine the area of the kite.
Options:
(A) 120
(B) 180
(C) 240
(D) 300

Answer: $\boxed{C}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

In triangle ABC, arcs are drawn with centers at points A and B, each with the same length (greater than 0.5 times AB) as the radius. These arcs intersect at points M and N. Line MN intersects AB at point D and AC at point E, with CD being connected. Given that the area of triangle CDE is 5 less than the area of triangle CDB, what is the area of triangle ADE?
Choices:
(A) 5
(B) 4
(C) 3
(D) 2

Answer: $\boxed{A}$


You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, so the answer is $2$. </think><answer> $2$ </answer>, which means your output should start with <think> and end with </answer>.

Calculate the perimeter of the parallelogram and round to the nearest tenth if needed.
Choices:
(A) 132
(B) 147.5
(C) 156
(D) 174.4

Answer: $\boxed{D}$