You are a helpful assistant good at solving math problems with step-by-step reasoning. You should first thinks about the reasoning process in the mind and then provides the user with the answer. Your answer must be in latex format and wrapped in $...$.The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> Since $1+1=2$, 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. Point P intersects circle O at point C, and points A and B are the points where PA and PB respectively are tangents to circle O. Given that the radius of circle O is 3cm and ∠APB = 60°. If the shaded sector in the diagram forms the lateral surface of a cone, what is the height of this cone?
Choices:
(A) 2√{2}cm
(B) √{2}cm
(C) √{10}cm
(D) \frac{3}{2}cm

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, with angle A measuring 32°. What is the degree measure of angle BOC?
Choices:
(A) 30°
(B) 64°
(C) 50°
(D) 28°

Answer: $\boxed{B}$


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

Determine the value of x.
Options:
(A) 10
(B) 15
(C) 20
(D) 25

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 ABE, angle E is 25°. The perpendicular bisector of AE, MN, intersects BE at point C. If AB is equal to AC, what is the degree measure of angle BAE?
Choices:
(A) 100°
(B) 105°
(C) 110°
(D) 120°

Answer: $\boxed{B}$


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

In the given diagram, AB is the diameter of circle O, C is an external point to circle O, a tangent is drawn from point C to circle O with point B as the point of tangency, and AC intersects circle O at point D. If ∠C = 40° and point E moves on the semicircle to the left of AB (not coinciding with A or B), what is the measure of ∠AED?
Choices:
(A) 20°
(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>.

If chords AC and DF are equidistant from the center of circle G with a radius of 26, what is the length of DE?
Choices:
(A) 12
(B) 20
(C) 24
(D) 26

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

Answer: $\boxed{D}$


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

In the given diagram, Xiaobin used a 50m long rope to enclose a parallelogram field, with one side measuring 16m. What is the length of the adjacent side?
Choices:
(A) 34m
(B) 18m
(C) 16m
(D) 9m

Answer: $\boxed{D}$


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

In the parallelogram ABCD shown in the figure, point E is the midpoint of side AD. Connecting EC intersects the diagonal BD at point F. If the area of triangle DEF is 5, what is the area of triangle BCF?
Choices:
(A) 10
(B) 15
(C) 20
(D) 25

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 isosceles triangles KLN and LMN, with angle JKN measuring 130 degrees, what is the measure of angle LKN?
Choices:
(A) 75
(B) 78
(C) 79
(D) 81

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 parallelogram WXYZ, what is the measure of angle XWZ?
Choices:
(A) 33
(B) 65
(C) 82
(D) 115

Answer: $\boxed{B}$


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

In the figure, quadrilateral ABDE is an inscribed quadrilateral of circle O, CE is the diameter of circle O, and BC, DC are connected. If ∠BDC = 20°, what is the degree measure of ∠A?
Choices:
(A) 90°
(B) 100°
(C) 110°
(D) 120°

Answer: $\boxed{C}$


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

In the given diagram, PA, PB, and CD are tangents to circle O, with points of tangency at A, B, and E respectively. CD intersects PA and PB at points C and D. If ∠APB = 60°, what is the measure of ∠COD?
(A) 50°
(B) 60°
(C) 70°
(D) 75°

Answer: $\boxed{B}$


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

In the given diagram, points A, B, C, and D are on the circle O, with angle ACB measuring 40° and angle CAB measuring 60°. What is the measure of angle ADC?
(A) 90°
(B) 100°
(C) 110°
(D) 120°

Answer: $\boxed{B}$


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

Determine the value of x.
Options:
(A) 2
(B) 3
(C) 4
(D) 5

Answer: $\boxed{A}$


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

In the figure, the bisectors of the two exterior angles ∠CBE and ∠CDF of quadrilateral ABCD intersect at point G. If ∠A = 52° and ∠DGB = 28°, what is the measure of ∠DCB?
Choices:
(A) 152°
(B) 128°
(C) 108°
(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 parallelogram ABCD, what is the measure of angle DAC?
Choices:
(A) 54
(B) 59
(C) 72
(D) 108

Answer: $\boxed{C}$


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

Calculate the area of the given figure. Which of the following options represents the correct area measurement?
(A) 98.4
(B) 112.5
(C) 114
(D) 128.1

Answer: $\boxed{B}$


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

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

Answer: $\boxed{A}$


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

In the given diagram, the height of the cylinder is 8 and the radius of the base is 3. If a plane is used to vertically cut the cylinder along the diameter of the top, what is the area of the resulting cross-section?
Choices:
(A) 24
(B) 48
(C) 32
(D) 72

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, PA and PB are tangents to circle O, with A and B as the points of tangency. AC is the diameter of the circle. If ∠CAB = 25°, what is the degree measure of ∠P?
Choices:
(A) 50°
(B) 65°
(C) 25°
(D) 75°

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, and lines CA, CB, DC, and DB are drawn. Given that ∠D = 30° and BC = 3, what is the length of AB?
Choices:
(A) 5
(B) 3√{2}
(C) 2√{3}
(D) 6

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 C, D are two points on circle O. If ∠BCD = 40°, what is the measure of ∠ABD?
Choices:
(A) 20°
(B) 40°
(C) 50°
(D) 60°

Answer: $\boxed{C}$


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

In the diagram, AD is the diameter of circle O, ∠{AB}=∠{CD}, and if ∠AOB=40°, what is the degree measure of the central angle ∠BPC?
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 figure, point P lies on segment AB, with PA=PB=PC=PD. If ∠BPC=60°, what is the measure of ∠BDC?
Choices:
(A) 15°
(B) 30°
(C) 25°
(D) 60°

Answer: $\boxed{B}$


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

In the given diagram, AB and AC are two chords of circle O, with angle A measuring 28°. The tangent passing through point C intersects the extension of OB at point D. What is the measure of angle D?
Choices:
(A) 28°
(B) 30°
(C) 32°
(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 given diagram, B and C are two points on circle A. The perpendicular bisector of AB intersects circle A at points E and F, and intersects segment AC at point D. If ∠BFC = 20°, what is the measure of ∠DBC?
(A) 30°
(B) 29°
(C) 28°
(D) 20°

Answer: $\boxed{A}$


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

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

Answer: $\boxed{D}$


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

In the figure, point O is a point inside triangle ABC, where OA=OB=OC, and ∠ABO=40°. What is the measure of ∠ACB?
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>.

Determine the scale factor from shape W to shape W'.
Choices:
(A) 2
(B) 4.5
(C) 9
(D) 18

Answer: $\boxed{A}$


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

Determine the value of the variable x rounded to the nearest tenth, assuming that segments that appear to be tangent are indeed tangent.
Choices:
(A) 14.1
(B) 14.6
(C) 15.1
(D) 15.6

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 triangle ABC, where AB = AC and CD is parallel to AB, point E lies on the extension of BC. If angle A is 30°, what is the measure of angle DCE?
Choices:
(A) 30°
(B) 52.5°
(C) 75°
(D) 85°

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 y that would make the quadrilateral a parallelogram.
Options:
(A) 8
(B) 9
(C) 25
(D) 96

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 5.
Options:
(A) 35
(B) 39
(C) 55
(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 figure, if AB is parallel to CD, CD is perpendicular to EF, and angle 1 is 125°, what is the measure of angle 2?
Choices:
(A) 25°
(B) 35°
(C) 55°
(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>.

What is the value of P multiplied by Q?
Choices:
(A) 12
(B) 15
(C) 16
(D) 18

Answer: $\boxed{B}$


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

In the given diagram, points A, B, C, and D are on circle O. Line AB intersects line DC at point E, and line AD intersects line BC at point F. If ∠E = 36° and ∠F = 30°, what is the measure of angle A in degrees?
Choices:
(A) 30°
(B) 40°
(C) 57°
(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>.

If a is parallel to b, what is the measure of angle 1?
(A) 47
(B) 54
(C) 79
(D) 101

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, with angle C measuring 20° and angle B measuring 50°. What is the measure of angle A?
Choices:
(A) 20°
(B) 25°
(C) 30°
(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>.

Calculate the perimeter of the triangle and round the answer to the nearest hundredth.
Choices:
(A) 27.31
(B) 28.52
(C) 60.34
(D) 61.55

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 figure, line a is parallel to line b, and the right-angled triangle ABC has its right angle vertex C on line b. If angle 1 is 54°, what is the degree measure of angle 2?
Choices:
(A) 36°
(B) 44°
(C) 46°
(D) 54°

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 Q R is parallel to T S, Q T is parallel to R S, and the measure of angle 1 is 131 degrees, what is the measure of angle 6 in the figure?
Choices:
(A) 101
(B) 111
(C) 121
(D) 131

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 E X equals 24 and D E equals 7, what is the value of Q X?
Choices:
(A) 7
(B) 18
(C) 24
(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 rectangle ABCD, where AB = 6 and AD = 8, P is a moving point on AD that is not coincident with A or D. Perpendicular lines are drawn through point P to intersect AC and BD at E and F, respectively. What is the value of PE + PF?
Choices:
(A) 10
(B) 4.8
(C) 6
(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>.

Determine the measurement of \angle 3 in the given figure.
Options:
(A) 10
(B) 15
(C) 20
(D) 25

Answer: $\boxed{D}$


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

In the given diagram, triangle ABC is folded along DE, resulting in point A lying outside quadrilateral BCDE. If angle 1 measures 70° and angle 2 measures 152°, what is the measure of angle A?
Choices:
(A) 40°
(B) 42°
(C) 30°
(D) 52°

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, AC is the diameter of circle O, triangle ABC is inscribed in circle O, AB = BC, ∠DBC = 32°. What is the measure of ∠BCD?
Choices:
(A) 113°
(B) 103°
(C) 45°
(D) 58°

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 and round it to the nearest tenth. Which of the following choices is the closest to the value of x?
(A) 28.7
(B) 38.0
(C) 43.7
(D) 50.3

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 angle CAB in rhombus ABCD measures 35 degrees, what is the measure of angle CDA?
Choices:
(A) 35
(B) 70
(C) 110
(D) 140

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, with circle O and radius OC perpendicular to chord AB, point D lies on minor arc AB (not coinciding with points A, B). If ∠AOC = 50°, what is the measure of ∠CDB?
(A) 25°
(B) 30°
(C) 40°
(D) 50°

Answer: $\boxed{A}$


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

Determine the value of x, rounding to the nearest tenth if needed. Assume that segments that seem to be tangent are indeed tangent.
Options:
(A) 5
(B) 10
(C) 15
(D) 20

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, PA, PB, and AB are all tangent to circle O, with angle P equal to 60°. What is the measure of angle AOB?
Choices:
(A) 50°
(B) 60°
(C) 70°
(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 the diameter of circle O, CD is a chord of circle O, and ∠ABD is 60°. What is the measure of angle C?
Choices:
(A) 15°
(B) 30°
(C) 45°
(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>.

Determine the value of x.
Options:
(A) 30
(B) 45
(C) 60
(D) 75

Answer: $\boxed{B}$


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

In triangle ABC, where angle C is 90 degrees, AC is equal to BC, AD bisects angle CAB and intersects BC at point D, DE is perpendicular to AB with E as the foot of the perpendicular, and AB is 6cm. What is the perimeter of triangle DEB?
(A) 4cm
(B) 6cm
(C) 8cm
(D) 10cm

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, the height of the cylinder is 8 and the radius of the base is 3. If a plane is used to vertically cut the cylinder along the diameter of the top, what is the area of the resulting cross-section?
Choices:
(A) 24
(B) 48
(C) 32
(D) 72

Answer: $\boxed{B}$


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

In the quadrilateral ABCD, where AB = 1, BC = 1, CD = 2, DA = √6, and ∠ABC = 90°, what is the area of quadrilateral ABCD?
Choices:
(A) 2
(B) 0.5 + √2
(C) 1 + √2
(D) (1 + √2) / 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>.

Calculate the area of the circle and round to the nearest tenth.
Choices:
(A) 18.8
(B) 37.7
(C) 113.1
(D) 452.4

Answer: $\boxed{C}$


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

In the figure, in circle O, the degree of angle AOB is 160°. C is a point on line AB, and D and E are two different points on line AB (not coinciding with points A and B). What is the degree of angle D plus the degree of angle E?
Choices:
(A) 160°
(B) 140°
(C) 100°
(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 right triangle ABC shown in the figure, with angle C equal to 90 degrees, BC equals 3, and AC equals 4, what is the value of sinA?
Choices:
(A) 3/4
(B) 4/3
(C) 3/5
(D) 4/5

Answer: $\boxed{C}$


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

In the given diagram, triangle ABC is inscribed in circle O, where AD is the diameter of circle O and angle ABC is 25°. What is the measure of angle CAD?
(A) 25°
(B) 60°
(C) 65°
(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, PA and PB are tangents to circle O, with A and B as the points of tangency. AC is the diameter of circle O, and ∠BAC is 35°. What is the degree measure of ∠P?
Choices:
(A) 35°
(B) 45°
(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 triangle ABC, as shown in the figure, DE is parallel to AB and FG, and the ratio of the distances from FG to DE and AB is 1:2. If the area of triangle ABC is 32 and the area of triangle CDE is 2, what is the area S of triangle CFG?
(A) 6
(B) 8
(C) 10
(D) 12

Answer: $\boxed{B}$


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

In the figure, in quadrilateral ABCD, EF passes through the intersection point O of the diagonals, intersects AD at E, and intersects BC at F. If AB = 4, BC = 5, and OE = 1.5, what is the perimeter of quadrilateral EFCD?
Choices:
(A) 16
(B) 14
(C) 12
(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>.

What is the value of x in the given figure?
Options:
(A) 10
(B) 12
(C) 14
(D) 15

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 parallelogram ABCD, where AB = 10, the angle bisector of ∠BAD intersects the extension of BC at point E and intersects DC at point F, with F being the midpoint of DC. DG is perpendicular to AE with G as the foot of the perpendicular. If DG = 3, what is the length of AE?
(A) 2√{3}
(B) 4
(C) 8
(D) 16

Answer: $\boxed{D}$


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

Determine the value of y that would make the quadrilateral a parallelogram.
Options:
(A) 16
(B) 32
(C) 74
(D) 80

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 cosine of angle A.
Choices:
(A) \frac { \sqrt { 26 } } { 26 }
(B) \frac { 2 \sqrt { 26 } } { 26 }
(C) \frac { 5 \sqrt { 26 } } { 26 }
(D) \frac { 7 \sqrt { 26 } } { 26 }

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>.

If the diameter of Circle A is DF and PG, and PA is equal to 7, what is the value of PG?
(A) 7
(B) 14
(C) 28
(D) 35

Answer: $\boxed{B}$


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

If x is less than 90 degrees in the given figure, which of the following statements must be true?
Choices:
(A) l is parallel to m
(B) l is perpendicular to m
(C) y is less than 90 degrees
(D) y is equal to 90 degrees
(E) y is greater than 90 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 figure, in quadrilateral ABCD, if angle B is 40° larger than angle A, what is the measure of angle D?
Choices:
(A) 60°
(B) 70°
(C) 100°
(D) 110°

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, if △AOC is congruent to △BOD, ∠C and ∠D are corresponding angles, and AC and BD are corresponding sides, with AD = 10cm, OD = OC = 2cm, what is the length of OB?
Choices:
(A) 8cm
(B) 10cm
(C) 2cm
(D) 0

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. The extension of CO intersects AB at point D, where BD equals BO, and angle A measures 50°. What is the measure of angle B?
Choices:
(A) 15°
(B) 20°
(C) 25°
(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>.

If a right-angled triangle ABC (with angles of 30° and 60°) and triangle CDE (with angles of 45° and 45°) are placed as shown in the figure, with the right angle vertex C overlapping, and DE parallel to BC, what is the degree measure of angle 1?
Choices:
(A) 75°
(B) 105°
(C) 110°
(D) 120°

Answer: $\boxed{B}$


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

In the figure, AB is a ladder leaning against a wall, with the foot of the ladder B 1.4m away from the wall, and point D on the ladder 1.2m away from the wall. If BD is 0.5m long, what is the length of the ladder?
Choices:
(A) 3.5m
(B) 3.85m
(C) 4m
(D) 4.2m

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 figure, AB is parallel to CD, BD is perpendicular to CF with B as the foot of the perpendicular, and ∠BDC is 55°. What is the degree measure of ∠ABF?
Choices:
(A) 55°
(B) 45°
(C) 35°
(D) 25°

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, DE is parallel to BC, AD/BD = 1/2, and BC = 12. What is the length of DE?
Choices:
(A) 3
(B) 4
(C) 5
(D) 6

Answer: $\boxed{B}$


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

In the given diagram, the measurement group at the school measured ∠BAC to be 90° at point A by the pond, and then measured ∠ACB to be 60° at point C, which is 10 meters away from point A. What is the distance between points A and B?
Choices:
(A) 10√{3}/3
(B) 10√{3}
(C) 5√{3}
(D) 20

Answer: $\boxed{B}$


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

In the given diagram, points A, B, and C are all on circle O. If ∠ACB = 20°, what is the measure of ∠AOB?
Choices:
(A) 20°
(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 triangle ABC, if angle A is 64° and the bisectors of angles ABC and ACD intersect at point A1, the bisectors of angles A1BC and A1CD intersect at point A2, and so on until the bisectors of angles A3BC and A3CD intersect at point A4, what is the measure of angle A4 in degrees?
(A) 5°
(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, ∠ACB is the central angle of arc AB in circle O. Point P is any point on the secant line m of circle O, and the midpoints of line segments PA and PB are D and E, respectively. If the radius of circle O is 8cm and ∠ACB = 30°, what is the length of line segment DE?
Choices:
(A) 2cm
(B) 4cm
(C) 4√{2}cm
(D) 4√{3}cm

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 with rectangle LMNP and parallelogram LKMJ, if the measure of angle MJN is 35 degrees, what is the measure of angle MPN?
Choices:
(A) 17.5
(B) 30
(C) 35
(D) 70

Answer: $\boxed{A}$


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

In the given diagram, a boat is sailing from east to west. At point A, the lighthouse M is observed in the direction of north by west 75°. After sailing 8 nautical miles to point B, the lighthouse M is observed in the direction of north by west 60°. What is the distance BM in nautical miles?
Choices:
(A) 8 nautical miles
(B) 4 nautical miles
(C) 4√{3} nautical miles
(D) 4√{2} nautical miles

Answer: $\boxed{A}$


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

In the figure, if the base radius of the cone is 2 and the slant height is 6, what is the lateral surface area?
Choices:
(A) 4π
(B) 6π
(C) 12π
(D) 16π

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, CD is a chord of circle O, and ∠ADC is 35°. What is the degree measure of ∠CBA?
Choices:
(A) 35°
(B) 45°
(C) 55°
(D) 65°

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, where AB = 8, BC = 10, the diagonals AC and BD intersect at point O. A line passing through point O intersects AD and BC at points E and F, respectively, with OE = 3. What is the perimeter of quadrilateral EFCD?
Choices:
(A) 20
(B) 24
(C) 28
(D) 32

Answer: $\boxed{B}$


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

In the given diagram, points A, B, and C lie on circle O, with angle AOB measuring 84°. What is the degree measure of angle ACB?
Choices:
(A) 40°
(B) 84°
(C) 42°
(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, AB is the diameter of circle O, CD is a chord, and AB is perpendicular to CD. If ∠BOC = 70°, what is the degree measure of ∠ABD?
Choices:
(A) 35°
(B) 45°
(C) 55°
(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 measure of angle AC in triangle ABC, where AB is congruent to BC, and ABC is an isosceles triangle.
Choices:
(A) 3
(B) 4
(C) 7
(D) 11

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 perimeter of triangle QRS if triangle QRS is similar to triangle QTP?
Choices:
(A) 9
(B) 14
(C) 21
(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 figure, A, B, and C are three points on circle O, with ∠ACB = 30°. What is the measure of ∠AOB?
Choices:
(A) 75°
(B) 60°
(C) 45°
(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>.

Determine the measure of angle D G if the measure of angle D G I is 24 degrees and the radius is 6 units.
Choices:
(A) 6
(B) \frac { 22 } { 5 } \pi
(C) 24
(D) 24 \pi

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, AB is the diameter of circle O, and CD is a chord of circle O. If ∠BAD = 56°, what is the measure of ∠ACD?
Choices:
(A) 34°
(B) 46°
(C) 56°
(D) 44°

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 a = 14, b = 48, and c = 50, what is the value of tangent B?
(A) 3.43
(B) 4.8
(C) 5.2
(D) 6.86

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, chord BC intersects radius OA at point D, and AB is connected to OC. If ∠A = 60° and ∠ADC = 85°, what is the degree measure of ∠C?
Choices:
(A) 35°
(B) 27.5°
(C) 30°
(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, when triangle ABC is folded along DE and point C falls outside quadrilateral ABDE, angle 1 measures 108° and angle C measures 35°. What is the measure of angle 2?
Choices:
(A) 35°
(B) 36°
(C) 37°
(D) 38°

Answer: $\boxed{D}$


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

In the diagram, pentagon ABCDE is inscribed in circle O. If ∠CAD = 40°, what is the sum of ∠B and ∠E in degrees?
Choices:
(A) 200°
(B) 215°
(C) 230°
(D) 220°

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>.

Find the value of w in parallelogram N Q R M.
Choices:
(A) 4
(B) 12
(C) 14
(D) 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>.

In triangle ABC, where angle A is 90°, point D lies on side AC, and DE is parallel to BC. If angle 1 is 155°, what is the measure of angle B?
(A) 45°
(B) 55°
(C) 65°
(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 figure, triangle ABC is similar to triangle DEF with point O as the center of similarity. Given that AB:DE = 0.3333333 and the perimeter of triangle ABC is 4, what is the perimeter of triangle DEF?
Choices:
(A) 8
(B) 12
(C) 16
(D) 36

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, PA intersects circle O at point A, PB intersects circle O at point B, and CD intersects circle O at point E. If ∠APB is 50°, what is the degree measure of ∠COD?
Choices:
(A) 50°
(B) 40°
(C) 25°
(D) 65°

Answer: $\boxed{D}$


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

In the figure shown, PA and PB are tangents to circle O at points A and B, with angle P measuring 70°. What is the measure of angle ACB?
Choices:
(A) 15°
(B) 40°
(C) 75°
(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>.

Calculate the area of the shaded sector and round to the nearest tenth.
Choices:
(A) 15.5
(B) 46.5
(C) 66.6
(D) 113.1

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 in the given figure.
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 diagram, O is the midpoint of the diagonal AC of rectangle ABCD, and M is the midpoint of side CD. If AB = 12 and OM = 4.5, what is the length of segment OB?
Choices:
(A) 7
(B) 8
(C) 7.5
(D) 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>.

Determine the value of z in the given figure.
Choices:
(A) 13.5
(B) 14
(C) 14.5
(D) 15

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 figure, it is known that line AB is parallel to CD, the bisector of angle GEB EF intersects CD at point F, and angle 1 is 42°. What is the measure of angle 2?
Choices:
(A) 138°
(B) 142°
(C) 148°
(D) 159°

Answer: $\boxed{D}$


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

In the diagram, AD is the bisector of ∠CAE, ∠B=45°, ∠DAE=75°, what is the measure of ∠ACD?
Choices:
(A) 105°
(B) 85°
(C) 60°
(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, AD is the bisector of ∠EAC, AD is parallel to BC, ∠B = 35°, what is the degree measure of ∠C?
Choices:
(A) 55°
(B) 45°
(C) 35°
(D) 25°

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 the sine of angle A.
Choices:
(A) \sqrt { 26 } / 26
(B) \sqrt { 26 } / 22
(C) \sqrt { 26 } / 5
(D) \sqrt { 26 } / 3

Answer: $\boxed{A}$


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

In the "Zhao Shuangxian diagram," a large square is composed of four congruent right-angled triangles and a small square. If the two legs of the right-angled triangles are 5 and 3 respectively, what is the area of the small square?
Choices:
(A) 4
(B) 3
(C) 2
(D) 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 figure, the diagonals of quadrilateral ABCD intersect at point O, and the sum of the lengths of the two diagonals is 36. If the perimeter of triangle OCD is 23, what is the length of AB?
Choices:
(A) 5
(B) 6
(C) 7
(D) 8

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 from the given choices:
(A) 4√2
(B) 4√3
(C) 4√6
(D) 8√6

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 the diameter of the circle, AB = √{2}AC, ∠ODC = 12°. What is the degree measure of ∠DBC?
Choices:
(A) 80°
(B) 102°
(C) 120°
(D) 135°

Answer: $\boxed{B}$


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

In the given diagram, if the diameter AB of circle O makes an angle of 31° with chord AC, and the tangent line PC passing through point C intersects the extension of AB at point P, what is the measure of angle ∠P?
Choices:
(A) 24°
(B) 25°
(C) 28°
(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 triangle ABC, angle A is 30 degrees and BC is 12. If BC is 12, what is the diameter of the circumscribed circle O?
Choices:
(A) 12
(B) 18
(C) 20
(D) 24

Answer: $\boxed{D}$


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

In the diagram, triangle ABC is an inscribed triangle in circle O, where AB is the diameter of circle O. Point D lies on circle O, and angle ADC is 65°. What is the measure of angle BAC?
Choices:
(A) 15°
(B) 25°
(C) 35°
(D) 45°

Answer: $\boxed{B}$


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

In the figure, quadrilateral ABCD is inscribed in circle O, E is a point on the extension of DC, and ∠A = 50°. What is the degree measure of ∠BCE?
Choices:
(A) 40°
(B) 50°
(C) 60°
(D) 130°

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, with points D and C on circle O. Lines AD, BD, DC, and AC are connected. If ∠BAD = 25°, what is the measure of angle C?
Choices:
(A) 75°
(B) 65°
(C) 60°
(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, AB is the diameter of circle O, points C and D are on circle O, and OD is perpendicular to AC. If ∠COB = 30°, what is the degree measure of ∠ACD?
Choices:
(A) 30°
(B) 37.5°
(C) 45°
(D) 60°

Answer: $\boxed{B}$


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

In the figure, triangle AOB is rotated 45° counterclockwise around point O to obtain triangle A'OB'. If angle AOB is 15°, what is the measure of angle AOB'?
Choices:
(A) 30°
(B) 40°
(C) 50°
(D) 60°

Answer: $\boxed{A}$


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

In triangle ABC, where AB = AC, points D and E are on BC and AC respectively. If AD = AE, ∠BAD = ∠CAD = 20°, what is the measure of ∠EDC?
(A) 30°
(B) 20°
(C) 10°
(D) 5°

Answer: $\boxed{C}$


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

In the given diagram, triangle ABC is inscribed in circle O, where AD is the diameter of circle O and angle ABC is 35°. What is the measure of angle CAD?
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, in quadrilateral ABCD, angle A is 70°. Quadrilateral ABCD is folded so that points D and C coincide with points F and E respectively (points F and E are both on the line AB). The fold line is MN. What is the measure of angle AMF?
(A) 70°
(B) 40°
(C) 30°
(D) 20°

Answer: $\boxed{B}$


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

In the given diagram, quadrilateral ABCD is an inscribed quadrilateral in circle O. The extensions of AD and BC intersect at point E, while the extensions of BA and CD intersect at point F. If ∠DCE = 80° and ∠F = 25°, what is the measure of angle E?
Choices:
(A) 55°
(B) 50°
(C) 45°
(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, the sum of the areas of squares 1 and 4 is 7, and the sum of the areas of squares 2 and 3 is 4. What is the sum of the areas of squares a, b, and c?
Choices:
(A) 10
(B) 13
(C) 15
(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 parallelogram ABCD, where AB = 10, the angle bisector of ∠BAD intersects the extension of BC at point E and intersects DC at point F, with F being the midpoint of DC. DG is perpendicular to AE with G as the foot of the perpendicular. If DG = 3, what is the length of AE?
(A) 2√{3}
(B) 4
(C) 8
(D) 16

Answer: $\boxed{D}$


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

In the given diagram, points A, B, C, and D are all on circle O, with angle B measuring 55°. If AO is parallel to DC, what is the degree measure of angle AOD?
Choices:
(A) 70°
(B) 75°
(C) 80°
(D) 85°

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>.

On a rectangular grass field with a length of 12m and a width of 6m, there is a curved asphalt path (with a constant width of 2m). What is the area of the grass field not covered by the path?
Choices:
(A) 70
(B) 60
(C) 48
(D) 18

Answer: $\boxed{B}$


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

In the given diagram, it is known that line AD is a tangent to circle O with point A as the point of tangency. Point D lies on circle O and intersects circle O at point B. Point C is on circle O, and angle ODA is 36°. What is the degree measure of angle ACB?
Choices:
(A) 54°
(B) 36°
(C) 30°
(D) 27°

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, AB is parallel to CD, ∠1 is 58°, FG bisects ∠EFD. What is the measure of ∠FGB in degrees?
Choices:
(A) 29°
(B) 151°
(C) 122°
(D) 109°

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, if the radius of circle O is 5 and the distance from point A to the center O is 3, what is the length of the shortest chord passing through point A?
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 circle R, if the measure of angle SU is 20 degrees, the measure of angle YW is 20 degrees, and the measure of angle YX is 45 degrees, what is the measure of angle ST?
(A) 20
(B) 30
(C) 45
(D) 90

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 AOB, with angle AOB measuring 120 degrees and BD and AC as the two altitudes, if AB equals 4, what is the length of DC?
Choices:
(A) √{3}
(B) 2
(C) \frac{3√{3}}{2}
(D) \frac{3√{3}}{4}

Answer: $\boxed{B}$


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

In the figure, AD, AE, and BC intersect ⊙O at points D, E, and F. If AD = 18, what is the perimeter of triangle ABC?
Choices:
(A) 18
(B) 27
(C) 36
(D) 54

Answer: $\boxed{C}$