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 points C and D are on circle O. If ∠ABC = 50°, what is the measure of ∠D?
Choices:
(A) 30°
(B) 40°
(C) 50°
(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 triangle ABC, where angle ACB is 90 degrees, AC is 3, and BC is 4. A circle is drawn with center A and tangent to BC. What is the radius of this circle?
Choices:
(A) 2.5
(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>.

If a point P outside the circle O is connected to the circle O with tangents PA and PB, with points A, B, and C as the points of tangency, and if ∠APB = 50°, what is the measure of ∠ACB?
Choices:
(A) 50°
(B) 60°
(C) 65°
(D) 70°

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 m is parallel to line n. If a triangle board ABC with a 45° angle at vertex C is placed on line n, then the sum of angles 1 and 2 is equal to which of the following?
(A) 30°
(B) 45°
(C) 60°
(D) 90°

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 quadrilateral ABCD inscribed in circle Z, angle BZA measures 104 degrees, angle CBA measures 94 degrees, and AB is parallel to DC. What is the measure of angle BZA?
Choices:
(A) 36
(B) 90
(C) 104
(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 two circles with O as the center, the radius of the larger circle is 5 and the radius of the smaller circle is 3. What is the chord length of the larger circle that is tangent to the smaller circle?
Choices:
(A) 4
(B) 6
(C) 8
(D) 10

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 circle O, AB is a diameter, and C, D are two points on the circle O with OC parallel to AD. If ∠DAC = 25°, what is the degree measure of ∠BOC?
Choices:
(A) 40°
(B) 50°
(C) 60°
(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, triangle ABC is an inscribed triangle in circle O, D is the midpoint of side AB, OA = 6, ∠ACB = 30°, then OD = ()
Choices:
(A) 6
(B) 3√{3}
(C) 3
(D) \frac{3√{3}}{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>.

Given that the measure of angle AOB is 40 degrees and the length of arc AB is 4π, what is the area of the sector AOB in the figure provided?
Choices:
(A) 4*pi
(B) 16*pi
(C) 36*pi
(D) 128*pi
(E) 324*pi

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, on a slope with an angle of 30°, the horizontal distance between two trees AC is 2m. What is the slope distance AB between the two trees?
Choices:
(A) 4m
(B) √{3}
(C) \frac{4√{3}}{3}m
(D) 4√{3}m

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 a certain flower nursery, there is an empty land in the shape of a trapezoid ABCD, with midpoints E, F, G, and H on each side. The diagonal AC measures 10 meters. If they want to enclose the quadrilateral EFGH with a fence, what is the total length of the fence needed?
Choices:
(A) 40 meters
(B) 30 meters
(C) 20 meters
(D) 10 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, point O is on side BC. Circle O with center O and radius BO is tangent to AC at point A. Point D is a point on arc AB where angle ADB is 60°. What is the degree measure of angle C?
Choices:
(A) 65°
(B) 50°
(C) 40°
(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, with angle A measuring 36°, AB equal to AC, BF bisecting angle ABC, E being the midpoint of AF, DE perpendicular to AC intersecting AB at D, and DC intersecting BF at P, what is the measure of angle DPB?
(A) 36°
(B) 54°
(C) 72°
(D) 90°

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 figure, line a is parallel to line b, angle 1 is 85°, and angle 2 is 35°. What is the degree measure of angle 3?
Choices:
(A) 40°
(B) 45°
(C) 50°
(D) 55°

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 circle B, D and C are diameters. Determine the measure of angle CD.
Choices:
(A) 35
(B) 45
(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, point C lies on circle O. If ∠ACB = 30°, what is the measure of ∠AOB?
Choices:
(A) 40°
(B) 60°
(C) 80°
(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>.

Determine the value of x from the following options:
(A) 4
(B) 6
(C) 8
(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>.

In the rhombus ABCD shown in the figure, where AB = 12, point E lies on AD, BE intersects AC at point F. If AF/FC = 1/3, what is the length of AE?
(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>.

In the figure, point P is inside triangle ABC, and D, E, F are the symmetric points of P with respect to AB, BC, AC respectively. If the internal angles of triangle ABC are ∠A=70°, ∠B=60°, ∠C=50°, then what is the sum of ∠ADB, ∠BEC, and ∠CFA?
Choices:
(A) 180°
(B) 270°
(C) 360°
(D) 480°

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 triangles FGH and FHJ inscribed in circle K, with angle FG congruent to angle FI. If the measure of angle 1 is 6x - 5, and the measure of angle 2 is 7x + 4, find the value of x.
Choices:
(A) 3
(B) 7
(C) 20
(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 the diagram, squares are constructed externally on each side of a right-angled triangle. The areas of two small squares are 9 and 25 respectively. What is the area of square A?
Choices:
(A) 16
(B) 32
(C) 34
(D) 64

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, AB is perpendicular to BD, ED is perpendicular to BD, C is the midpoint of segment BD, and AC is perpendicular to CE. If ED = 1 and BD = 4, what is the value of AB?
Choices:
(A) 2
(B) 3
(C) 4
(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>.

Determine the measure of angle U.
Choices:
(A) 60
(B) 68
(C) 76
(D) 136

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>.

Given that MC = 7, RM = 4, and AT = 16, what is the value of SC?
(A) 10.5
(B) 12
(C) 13.5
(D) 21

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, C is a point on circle O, AD is perpendicular to the tangent passing through point C, with D as the foot of the perpendicular. If ∠DCA = 55°, what is the degree measure of ∠CAO?
Choices:
(A) 25°
(B) 35°
(C) 45°
(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>.

In the diagram, BC is the diameter of circle O, and A and D are two points on circle O. If ∠D = 35°, what is the degree measure of ∠OAC?
Choices:
(A) 35°
(B) 55°
(C) 65°
(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 figure, given that AB is parallel to DE, ∠ABC = 70°, and ∠CDE = 140°, what is the value of ∠BCD?
Choices:
(A) 20°
(B) 30°
(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>.

What are the values of X and Y?
Choices:
(A) 3.5
(B) 7
(C) 34
(D) 41

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, point C is an external point to circle O, CA and CD are tangents to circle O with A and D as points of tangency, and BD and AD are connected. If ∠ACD = 48°, what is the measure of ∠DBA?
Choices:
(A) 48°
(B) 60°
(C) 66°
(D) 32°

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>.

Given that line BE is perpendicular to line AD and line CF is also perpendicular to line AD, with AE being equal to EF, what is the value of x in the figure above?
Choices:
(A) 40
(B) 45
(C) 50
(D) 55
(E) 60

Answer: $\boxed{E}$


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 PQR, given that ZQ = 3a - 11, ZP = a + 5, PY = 2c - 1, YR = 4c - 11, angle PRZ = 4b - 17, angle ZRQ = 3b - 4, angle QYR = 7b + 6, and angle PXR = 2a + 10. PX is an altitude of triangle PQR. Determine the value of a.
Options:
(A) 10
(B) 20
(C) 30
(D) 40

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 two tangents to circle O, with points of tangency at A and B. If ∠APO = 25°, what is the measure of ∠AOB?
(A) 140°
(B) 130°
(C) 120°
(D) 110°

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, rectangle ABCD and triangle BDE are given, with point A lying on BE. If the area of rectangle ABCD is 20 and the area of triangle BDE is 24, what is the area of triangle ADE?
Choices:
(A) 10
(B) 12
(C) 14
(D) 16

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, points PA and PB are tangents to circle O at points A and B respectively. Point C is a moving point on the minor arc AB (not coinciding with A or B), and angle P is 70°. What is the measure of angle C?
Choices:
(A) 110°
(B) 115°
(C) 120°
(D) 125°

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 scenario, there is an electric pole AB and a square billboard (ignoring the thickness of the sign) on the roadside. One day, Xiaoming noticed that under the sunlight, the shadow of the top end A of the electric pole falls exactly on the midpoint G of the upper side of the square billboard, while the shadow of the square billboard falls exactly on point E on the ground. Given that BC=5 meters, the side length of the square is 2 meters, and DE=4 meters, what is the height of the electric pole at this moment in meters?
Choices:
(A) 8
(B) 7
(C) 6
(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 given diagram of two congruent triangles, where the letters represent the side lengths of the triangles, what is the degree measure of ∠1?
Choices:
(A) 62°
(B) 72°
(C) 76°
(D) 66°

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, P is a point outside the circle O, and PA is a tangent to circle O. If PO = 13 and PA = 12, what are the area and circumference of circle O?
Choices:
(A) 5π, 10π
(B) 10π, 5π
(C) 25π, 10π
(D) 10π, 25π

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>.

What is the value of X in equation A?
Choices:
(A) 2
(B) 4
(C) 8
(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, line segment AB is the diameter of circle O, chord CD is perpendicular to AB, and angle CAB is 40°. What are the measures of angles ABD and AOD, respectively?
(A) 40°
(B) 50°
(C) 80°
(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>.

In the right triangle ABC, where angle C is a right angle and CD is perpendicular to AB at D, given that AC = 4 and AB = 5, what is the value of tan angle BCD?
(A) 3/4
(B) 4/3
(C) 3/5
(D) 4/5

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>.

In triangle ABE, where BA=BE and F is the midpoint of AE, if ∠ABC=34° and ∠C=50°, what is the measure of ∠ADB?
(A) 60°
(B) 63°
(C) 67°
(D) 70°

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 following options:
(A) 46.5
(B) 87
(C) 93
(D) 105

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>.

Find the value of z in parallelogram M N P R.
Choices:
(A) 6.1
(B) 11.1
(C) 15.3
(D) 18.3

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>.

Given that each pair of polygons is similar, what is the value of y? The possible choices are (A) 7, (B) 8, (C) 10, and (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>.

Determine the value of y.
Options:
(A) 1
(B) 3
(C) 5
(D) 8

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 diagonal of rectangle ABCD intersects at point O. If ∠ACB = 30°, AB = 2, what is the length of OC?
Choices:
(A) 2
(B) 3
(C) 2√{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 diagram, there is a parallelogram ABCD and a square CEFG, with point E on AD. If ∠ECD = 35° and ∠AEF = 15°, what is the degree measure of ∠B?
Choices:
(A) 50
(B) 55
(C) 70
(D) 75

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, AB is the diameter of the semicircle O, C and D are two points on the semicircle, and D is the midpoint of segment AC. If ∠B = 70°, what is the degree measure of ∠DAB?
Choices:
(A) 54°
(B) 55°
(C) 56°
(D) 57°

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 AC = 4cm, the perpendicular bisector of segment AB intersects AC at point N. If the perimeter of triangle BCN is 7cm, what is the length of BC?
Choices:
(A) 3
(B) 4
(C) 7
(D) 11

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, quadrilateral ABCD and quadrilateral A'B'C'D' are similar figures with point O as the center of similarity. If OA:OA' = 2:3 and the area of quadrilateral ABCD is 4, what is the area of quadrilateral A'B'C'D'?
Choices:
(A) 3
(B) 4
(C) 6
(D) 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 triangle ACE, point D lies on side AC and point B lies on the extension of side CE. If angle A is 47°, angle B is 55°, and angle C is 43°, what is the measure of angle DFE?
Choices:
(A) 125°
(B) 45°
(C) 135°
(D) 145°

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 are on circle O. Quadrilateral ABCD is a parallelogram with CD tangent to circle O. Line AD intersects circle O at point E, and angle D measures 70°. What is the measure of angle BEC?
Choices:
(A) 50°
(B) 60°
(C) 70°
(D) 80°

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 an inscribed quadrilateral in circle O. If ∠BOD = 90°, what is the measure of ∠BCD?
Choices:
(A) 90°
(B) 125°
(C) 135°
(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>.

In the figure, D and E are the midpoints of AB and AC, respectively. F is a point on DE, and ∠AFB = 90°. If AB = 6 and BC = 8, what is the length of EF?
Choices:
(A) 1
(B) 2
(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>.

Determine the measure of angle 1.
Options:
(A) 31
(B) 42
(C) 59
(D) 63

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, D and E are the midpoints of BC and AC, respectively. BF bisects angle ABC. Point F intersects DE. If AB = 8 and BC = 6, what is the length of EF?
Choices:
(A) 1
(B) 2
(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 figure, in circle O, OA is perpendicular to BC, and ∠AOC is 50°. What is the degree measure of ∠ADB?
Choices:
(A) 15°
(B) 25°
(C) 30°
(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>.

If polygon BCDE is congruent to polygon RSTU, what is the value of w?
Choices:
(A) 3
(B) 10
(C) 20
(D) 42

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, the chords AB and CD intersect at point P, and AC and BD are connected. If angle A is 35° and angle BPC is 78°, what is the measure of angle B?
Choices:
(A) 53°
(B) 43°
(C) 39°
(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>.

In the figure, AB is parallel to CD, OH intersects AB and CD at points F and H respectively, OG intersects AB and CD at points E and G respectively. If OE/EG = 4/9 and OF = 12, what is the length of OH?
Choices:
(A) 39
(B) 27
(C) 12
(D) 26

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, △ABC is congruent to △DCB, with corresponding vertices A and B matching with D and C. If AB = 7cm, BC = 12cm, and AC = 9cm, what is the length of BD?
Choices:
(A) 7cm
(B) 9cm
(C) 12cm
(D) Cannot be determined

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 measurement of angle 1.
Options:
(A) 33
(B) 45
(C) 54
(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, the diagonals of quadrilateral ABCD intersect at point O, with AD ≠ CD. A line OM is drawn perpendicular to AC passing through point O, intersecting AD at point M. If the perimeter of triangle CDM is 40cm, what is the perimeter of parallelogram ABCD?
Choices:
(A) 40cm
(B) 60cm
(C) 70cm
(D) 80cm

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, the perpendicular bisectors of AB and AC intersect BC at D and E respectively. If angle BAC is 110 degrees, what is the measure of angle DAE?
Choices:
(A) 40
(B) 45
(C) 50
(D) 55

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, OA is the radius of circle O, chord BC is perpendicular to OA, and D is a point on circle O. If ∠ADB = 28°, what is the degree measure of ∠AOC?
Choices:
(A) 14°
(B) 28°
(C) 56°
(D) 84°

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 AD is the diameter of circle O, AB is a chord of circle O, and ∠DAB = 50°, with point C on the circle, what is the degree measure of ∠ACB?
Choices:
(A) 100°
(B) 50°
(C) 40°
(D) 20°

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, points A, B, and C lie on circle O. The extension of CO intersects AB at point D. If ∠B = 40° and ∠ADC = 110°, what is the measure of angle A?
Choices:
(A) 50°
(B) 55°
(C) 60°
(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 given figure, if a is parallel to b and angle 1 measures 130°, what is the measure of angle 2?
Choices:
(A) 50°
(B) 70°
(C) 120°
(D) 130°

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, chords AC and BC are two chords, with ∠C = 35°. What is the degree measure of ∠O?
Choices:
(A) 25°
(B) 35°
(C) 65°
(D) 70°

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 the area of a circle is 20π square centimeters, what is the circumference of the circle?
Choices:
(A) \sqrt 5 \pi
(B) 2 \sqrt 5 \pi
(C) 4 \sqrt 5 \pi
(D) 20 \pi

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>.

Given that in the diagram, BA is equal to BC and x is equal to 117, what is the value of y?
Choices:
(A) 117
(B) 110
(C) 63
(D) 27

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, AB is the diameter of circle O, and C, D are two points on circle O. If ∠D = 35°, what is the degree measure of ∠OCB?
Choices:
(A) 35°
(B) 55°
(C) 65°
(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, AB is a chord of circle O, OC is perpendicular to AB, intersecting circle O at point C. If ∠ABC = 20°, what is the measure of ∠BAO in degrees?
Choices:
(A) 40°
(B) 45°
(C) 50°
(D) 55°

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, to measure the height of object AB, the angle of elevation at point D to point A is 30°. Moving 20 meters towards object AB direction to point C, the angle of elevation at point A is measured to be 60°. What is the height of object AB?
Choices:
(A) 10√{3} meters
(B) 10 meters
(C) 20√{3} meters
(D) \frac{20√{3}}{3} 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>.

In the given diagram, AB is a tangent to circle O, with point B as the point of tangency. The extension of AO intersects circle O at point C. If ∠A = 45° and AB = √2, what is the value of AC?
Choices:
(A) 2√2 + 1
(B) 4
(C) 2√2
(D) 2 + √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>.

What possible values could x take in the given diagram?
Choices:
(A) 10
(B) 20
(C) 40
(D) 50
(E) any of the above

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 y.
Options:
(A) 2.2
(B) 3
(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>.

In the diagram, points E and F are the midpoints of BD and CE, respectively. If the area of triangle AEF is 3, what is the area of triangle ABC?
Choices:
(A) 9
(B) 10
(C) 11
(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, line a is parallel to line b, and the right angle vertex C of triangle ABC lies on line b. If ∠1 is 26°, what is the measure of ∠2?
Choices:
(A) 26°
(B) 54°
(C) 64°
(D) 66°

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, assuming that segments that appear to be tangent are indeed tangent.
Options:
(A) 14
(B) 15
(C) 16
(D) 17

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, line a is parallel to line b, and the right angle vertex C of triangle ABC with a 30° angle is placed on line a. If ∠1 is 65°, what is the degree measure of ∠2?
Choices:
(A) 25°
(B) 30°
(C) 35°
(D) 40°

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, circle O is the circumcircle of triangle ABC, with angle A measuring 50°. What is the measure of angle OCB?
Choices:
(A) 60°
(B) 50°
(C) 40°
(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>.

In the given figure, the two chords AB and CD of the circle intersect at point E, with AD equal to CB, and angle A is 40°. What is the degree measure of angle CEB?
Choices:
(A) 50°
(B) 80°
(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 diagram, quadrilateral ABCD is inscribed in circle O. If one of its exterior angles ∠DCE is 65° and ∠ABC is 68°, what is the degree measure of ∠A?
Choices:
(A) 112°
(B) 68°
(C) 65°
(D) 52°

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, △ABC is similar to △A'B'C', AD and BE are the altitude and median of △ABC, A'D' and B'E' are the altitude and median of △A'B'C', and AD = 4, A'D' = 3, BE = 6. What is the length of B'E'?
Choices:
(A) \frac{3}{2}
(B) \frac{5}{2}
(C) \frac{7}{2}
(D) \frac{9}{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) 6
(B) 2√21
(C) 10
(D) 2√29

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 the figure is formed by the transformation of "Zhao Shuangxian", consisting of eight congruent right-angled triangles, and the areas of the squares ABCD, EFGH, and MNPQ are denoted as S1, S2, and S3 respectively, and if S1 + S2 + S3 = 45, what is the value of S2?
Choices:
(A) 12
(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>.

In the given diagram, AD is a tangent to circle O, with D as the point of tangency. Point D intersects circle O at point C, where CD is parallel to OA, and CB is connected. If ∠A = 50°, what is the measure of ∠OBC?
Choices:
(A) 40°
(B) 30°
(C) 25°
(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 the given diagram, AB is the diameter of circle O, and DC is a chord. If ∠BDC = 31°, what is the degree measure of ∠COB?
Choices:
(A) 64°
(B) 60°
(C) 62°
(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 triangle ABC, where angle C is 90 degrees, AB is 5, AC is 4, AD bisects angle BAC, DE is perpendicular to AB at E. If point P is a moving point on line AD, what is the minimum value of PE + PB?
Choices:
(A) 5
(B) 4
(C) 3.5
(D) 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 given diagram, AB is the diameter of circle O, DC is a chord. If ∠COB = 68°, what is the degree measure of ∠BDC?
Choices:
(A) 30°
(B) 32°
(C) 34°
(D) 45°

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, circle O is the circumcircle of triangle ABC. If angle OBA is 50°, what is the measure of angle C?
Choices:
(A) 30°
(B) 40°
(C) 50°
(D) 80°

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, lines PQ and PS are tangents to circle O, and chord QS is drawn. If angle P measures 40 degrees, what is the measure of angle PQS?
Choices:
(A) 140 degrees
(B) 80 degrees
(C) 70 degrees
(D) 60 degrees

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 CD is the diameter of circle O, and the chord DE passing through point D is parallel to radius OA. If the angle measure of ∠D is 50°, what is the degree measure of ∠C?
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 figure, given that line a is parallel to lines b and c, and lines m and n intersect lines a, b, and c at points A, C, E, B, D, F respectively, with AC = 3, CE = 6, and BD = 2, what is the length of DF?
(A) 4
(B) 4.5
(C) 3
(D) 3.5

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 y in the given figure.
Options:
(A) 13
(B) 13.5
(C) 14
(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>.

In triangle ABC, where angle C is 90 degrees, AC is 8, and BC is 6, point P is a moving point on the hypotenuse AB. Perpendiculars PE and PF are drawn from point P to AC and BC respectively, forming line segment EF. What is the minimum value of line segment EF?
Choices:
(A) 1.2
(B) 2.4
(C) 2.5
(D) 4.8

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, a car travels up a ramp at an angle of inclination α for 13 meters. Given that sinα = 5/13, what is the height the car ascends to?
Choices:
(A) 5 meters
(B) 6 meters
(C) 6.5 meters
(D) 7 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>.

In the figure, the quadrilateral paper ABCD is folded along MN, with points A and D landing at points A1 and D1 respectively. If ∠1 + ∠2 = 130°, what is the value of ∠B + ∠C?
Choices:
(A) 115°
(B) 130°
(C) 135°
(D) 150°

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 lie on the same circle, with ∠ABD=\frac{1}{3}∠ABC and ∠ACD=20°. What is the measure of angle DAC?
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 given figure, line AB is parallel to CD, angle 1 measures 50°, and angle 2 measures 110°. What is the measure of angle E?
Choices:
(A) 40°
(B) 50°
(C) 60°
(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>.

In the given diagram, two vertices of a right-angled triangle with a 30° angle are placed on the opposite sides of a ruler. If ∠1 = 22°, what is the degree measure of ∠2?
Choices:
(A) 22°
(B) 32°
(C) 38°
(D) 44°

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 circle with radius 1, where AP is a tangent to the circle with A as the point of tangency, and OP intersects chord AB at point C, where C is the midpoint of AB, and angle P is 30 degrees, what is the length of CP?
Choices:
(A) 2
(B) 1.5
(C) 1.6
(D) 1.8

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 A are on circle O, with ∠CAB = 20°. A tangent to circle O is drawn passing through point C and intersecting the extension of AB at point D. What is the measure of angle D?
Choices:
(A) 40°
(B) 50°
(C) 60°
(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 figure where line AB is parallel to line CD, and line EF intersects lines AB and CD at points G and H, if angle 1 is 105°, what is the measure of angle 2?
(A) 45°
(B) 55°
(C) 65°
(D) 75°

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 and round the side measurement to the nearest tenth.
Options:
(A) 12.7
(B) 15.2
(C) 28.9
(D) 34.8

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>.

What is the length of the base of triangle W X Y, which has an area of 248 square inches?
Choices:
(A) 15.5
(B) 16
(C) 31
(D) 32

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, AB is the diameter of circle O, point C is a point on circle O, and a tangent to circle O is drawn passing through point C and intersecting the extension of diameter AB at point D. If ∠A = 25°, what is the measure of ∠D?
(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, the angle bisector of ∠ABC intersects diagonal AC at point E and intersects CD at point M. Given that BC = 2, DM = 3, and the area of quadrilateral ABCD is 28, what is the area of triangle ABE?
Choices:
(A) \frac{28}{3}
(B) \frac{21}{2}
(C) 10
(D) \frac{14}{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>.

Given that each pair of polygons is similar, what is the value of C D?
(A) 3.333
(B) 10/3
(C) 4
(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>.

In the figure, M is the midpoint of side BC of triangle ABC, AN is the exterior angle bisector of triangle ABC, BN is perpendicular to AN at point N, and AB = 4, MN = 2.8. What is the length of AC?
Choices:
(A) 1.2
(B) 1.4
(C) 1.6
(D) 1.8

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, if ∠1 = 125°, what is the sum of ∠1, ∠2, ∠M, ∠N, ∠P, and ∠Q?
Choices:
(A) 25°
(B) 375°
(C) 360°
(D) 385°

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, a circle with center O on point A of AB and radius OA is tangent to BC at point C. If angle A is 28°, what is the measure of angle B?
Choices:
(A) 28°
(B) 56°
(C) 44°
(D) 34°

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 right triangle Rt△AOB shown in the figure, where OA=OB=5 and the radius of circle ⊙O is 3, point P is a moving point on side AB. A tangent line PQ is drawn passing through point P to circle ⊙O (with point Q as the point of tangency). What is the minimum value of the tangent line PQ?
Choices:
(A) 3
(B) 5√{2}-3
(C) 4
(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>.

Determine the value of x and round the side measurement to the nearest tenth.
Options:
(A) 4.1
(B) 4.8
(C) 5.3
(D) 6.1

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 parallelogram ABCD shown in the figure, where AE is perpendicular to BC at E, AF is perpendicular to CD at F, and ∠ABC = 75°, what is the degree measure of ∠EAF?
Choices:
(A) 60°
(B) 65°
(C) 70°
(D) 75°

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, PA, PB, and AB are all tangent to circle O, with angle P measuring 40°. What is the measure of angle AOB?
Choices:
(A) 40°
(B) 50°
(C) 60°
(D) 70°

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 circle with radius 1, ⊙O, shown in the figure, AP is a tangent to ⊙O with A as the point of tangency. OP intersects chord AB at point C, where C is the midpoint of AB. If ∠P = 30°, what is the length of CP?
Choices:
(A) 2
(B) 1.5
(C) 1.6
(D) 1.8

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 measure of angle J.
Choices:
(A) 102
(B) 107
(C) 112
(D) 117

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, triangle ABC is an inscribed triangle of circle O, where BD is the diameter of circle O. If angle ABD is 20°, what is the measure of angle ACB?
Choices:
(A) 70°
(B) 65°
(C) 60°
(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 figure, quadrilateral ABCD is folded along the line EF. Given that ∠A = 110° and ∠B = 80°, what is the sum of angles 1 and 2?
Choices:
(A) 20°
(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 measurement of angle 4.
Options:
(A) 38
(B) 42
(C) 52
(D) 62

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, AC and BD are the diagonals of the rhombus ABCD, with BH perpendicular to AD at point H. If AC is 4 and BD is 3, what is the length of BH?
Choices:
(A) 2.4
(B) 2.5
(C) 4.8
(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>.

In the given diagram, AB is the diameter of circle O. Points C and D are on circle O, with ∠CDB = 25°. A tangent to circle O passing through point C intersects the extension of AB at point E. What is the measure of angle ∠E?
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, point O is the center of the circumcircle of triangle ABC, and OB is connected. If angle 1 is 37°, what is the measure of angle 2?
Choices:
(A) 52°
(B) 51°
(C) 53°
(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>.

What is the measure of angle Z X V?
Choices:
(A) 25
(B) 65
(C) 115
(D) 130

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 showing the construction of line PS parallel to line RQ through point P, which statement provides the justification for this construction?
Choices:
(A) angle 1 = angle 2
(B) angle 1 = angle 3
(C) line PR = line RQ
(D) line PS = line RQ

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, points A, B, and C lie on circle O. If ∠AOB = 70°, what is the measure of ∠ACB?
Choices:
(A) 20°
(B) 35°
(C) 45°
(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>.

Determine the area of a regular hexagon with a perimeter of 72 feet.
Choices:
(A) 187.1
(B) 305.5
(C) 374.1
(D) 532

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>.

Given that points S, T, and U are the midpoints of line segments DE, EF, and DF, respectively, what is the value of z?
Choices:
(A) 0.383
(B) 0.575
(C) 1.15
(D) 2.875

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 if the segment is tangent to the circle.
Options:
(A) 12
(B) 16
(C) 20
(D) 22

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, BD is the diameter of circle O, and ∠A is 60°. What is the degree measure of ∠DBC?
Choices:
(A) 30°
(B) 45°
(C) 60°
(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 figure, in circle O, diameter AB intersects chord CD at point E. Connect BC, AD, and the tangent passing through point C intersects the extension of AB at point F. If ∠D = 65°, what is the degree measure of ∠F?
(A) 30°
(B) 35°
(C) 40°
(D) 45°

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.
Options:
(A) 6
(B) 8
(C) 10
(D) 12

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, and C lie on circle O. If angle A measures 66°, what is the measure of angle OCB?
Choices:
(A) 48°
(B) 33°
(C) 28°
(D) 24°

Answer: $\boxed{D}$