ydeng9/OpenVLThinker-grpo-hard · Datasets at Hugging Face

problem stringlengths 7 550	answer stringlengths 1 22	source stringclasses 3 values
<image>As shown in the picture, Xiaobin used a 50m long rope to enclose a parallelogram field. If one side is 16m long, then its adjacent side is ().	9m	geoqa
<image>Find m \angle 2.	24	geoqa
<image>In the provided diagram, parallelogram ABCD is shown where AB measures 5.0. The diagonals intersect at point O. Given that the perimeter of triangle OCD is 23.0, what is the sum of the lengths of the two diagonals of parallelogram ABCD?	36	geoqa
<image>If points C and D are situated on circle O with AB serving as the diameter, and angle ACD is measured at 15.0 degrees, what is the degree measure of angle BAD?	75°	geoqa
<image>Use parallelogram A B C D to find m \angle A F B.	131	geoqa
<image>Circle A has diameters D F and P G. If P A = 7, find P G.	14	geoqa
<image>In the provided figure, circle O has points A, B, and C. If angle AOB measures 70.0 degrees, what is the degree measure of angle ACB?	35°	geoqa
<image>In the provided figure, points A, B, C, and D lie on circle O, with BC being the diameter of circle O. If the measure of angle D is 36.0 degrees, what is the degree measure of angle BCA?	54°	geoqa
<image>What is the measure of angle 1 in triangle ABC, where angle C is 90 degrees, EF is parallel to AB, and angle B measures 39 degrees?	51°	geoqa
<image>In the given figure showing parallelogram ABCD, point P lies on side AD such that the ratio of AP to DP is 1.0:2.0. If the shaded region in the figure has an area of 3.0, what is the area of parallelogram ABCD?	9	geoqa
<image>Find the area of the figure. Round to the nearest tenth if necessary.	374.5	geoqa
<image>As shown in the figure, in the parallelogram ABCD, AB = 4.0, the bisector of angle BAD and the extended line of BC intersect at point E, and DC at point F, and DF: CF = 3.0, DG perpendicular AE, and the vertical foot is G , If DG = 1.0, then the length of AE is ()	\frac{16√{2}}{3}	geoqa
<image>In the provided diagram, points A, B, and E are all in a straight line. The square ABCD has a side length of 6 units, while the square BEFG has a side length of 8 units. Point H is the midpoint of the line segment DF. What is the length of line segment BH?	5√{2}	geoqa
<image>Find the area of the figure. Round to the nearest tenth if necessary.	156	geoqa
<image>In the provided figure of quadrilateral ABCD, where AB measures 4 units, BC measures 6 units, and AE measures 2 units, what is the length of AF, the altitude on side DC?	3	geoqa
<image>What is the measure of angle A if angle B is 60.0 degrees, angle AED is 45.0 degrees, and DE is parallel to BC in the given figure?	75°	geoqa
<image>In the given figure, line a is perpendicular to line c and line b is also perpendicular to line c. If angle 1 measures 70.0 degrees, what is the measure of angle 2?	70°	geoqa
<image>What is the measure of angle ACB if angles APB and APB both touch circle O and have a measure of 50.0 degrees, and point C lies on circle O?	65°	geoqa
<image>In the given figure, line AB is parallel to line CD. Additionally, the bisector EF of angle BEG intersects line CD at point F. If angle 1 measures 42.0, what is the measure of angle 2?	159°	geoqa
<image>As shown in the figure, in triangle ABC, point M is the midpoint of BC, AD is the bisector of exterior angle of triangle ABC, and AD perpendicular BD, if AB = 6.0, AC = 9.0, the length of MD is ()	\frac{15}{2}	geoqa
<image>In the given figure, circle O is depicted with chords AB and CD intersecting at point P. If the measure of angle A is 42.0 degrees and angle APD is 77.0 degrees, what is the measure of angle B?	35°	geoqa
<image>Consider the parallelogram ABCD with angle B measuring 110.0 degrees. If AD is extended to point F and CD is extended to point E, resulting in the connection of EF, what is the sum of angles E and F? Denote the sum as 'θ'.	70°	geoqa
<image>A B C D is a trapezoid. If A C = 3 x - 7 and B D = 2 x + 8, find the value of x so that A B C D is isosceles.	15	geoqa
<image>In the given diagram of triangle ABC, where EF is parallel to BC, and the ratio of AE to BE is 1:2, with EF having a length of 3.0 units, what is the value of BC?	9	geoqa
<image>What is the measure of angle BCF if, in the provided figure, AD is equal to BC and AE is equal to CF, E and F are points on BD, and BE is equal to DF, with angle AEB measuring 100 degrees and angle ADB measuring 30 degrees?	70°	geoqa
<image>In triangle ABC, points D and E lie on AB and AC respectively. DE is parallel to BC. If AD measures 5.0, BD measures 10.0, and AE measures 3.0, what is the length of AC?	9	geoqa
<image>In the given triangle ABC, with angle ACB measuring 90°, three squares are constructed outside the triangle using each side of the triangle. If the areas of these squares are represented by S1, S2, and S3, and S1 equals 3 while S2 equals 11, what is the value of S3?	8	geoqa
<image>Triangle J K L is circumscribed about \odot R. Find the perimeter of \triangle J K L.	52	geoqa
<image>P Q is tangent to the circle. Find x. Round to the nearest tenth.	5.2	geoqa
<image>In the given triangle ABC, points D and E are situated on the sides AC and BC respectively, with DE being parallel to AB. If the ratio of CD to CA is 2:3 and the area of triangle ABC is 18.0, what is the area of triangle DEC?	8	geoqa
<image>In the given triangle ABC, the midpoints of BC and AC are denoted by D and E, respectively. The angle ABC is bisected by BF, which intersects DE at point F. If the length of BC is 6 units, what is the length of DF?	3	geoqa
<image>In the provided figure, the ray OA has its endpoint O lying on the line CD. If angle COA is measured at 37.0 degrees, what is the measure of angle AOD?	143°	geoqa
<image>In the given diagram, circle O is depicted and AB is the diameter of circle O. The length of AB is equal to the square root of 2 times the length of AC. If angle ODC measures 12 degrees, what is the measure of angle DBC?	102°	geoqa
<image>In a garden, there is a pentagonal open space as depicted in the figure. To enhance the aesthetics, there is a plan to plant flowers and plants at each vertex of the pentagon, with each plant placed at the center of a circle with a radius of 2.0 units. What is the total area of the sector formed by each circle? Represent the total area as 'A'.	6πm²	geoqa
<image>In the given diagram, triangle ABC is congruent to triangle ADE. If the measure of angle BAE is 135° and the measure of angle DAC is 55°, what is the measure of angle CFE?	40°	geoqa
<image>In the provided figure, circle O has diameter AB, and chord CD is perpendicular to AB. Point E is the foot of the perpendicular. Point G can be anywhere on arc AC. By extending the line AG and DC to point F, and connecting GC, GD, and AD, what is the measure of angle AGD if angle BAD is 25.0 degrees?	65°	geoqa
<image>In the given diagram, if the line segment AB is the diameter of circle O and chord CD is perpendicular to AB, with angle CAB measuring 20.0, what is the measure of angle AOD?	140°	geoqa
<image>In the given figure, AB represents the diameter of circle O, while AC denotes the tangent of circle O, with point D where OC connects to circle O. If angle C is 36.0 degrees, what is the measure of angle B?	27°	geoqa
<image>In the given triangle ABC, with the angle bisector of angle ABC intersecting the perpendicular bisector of AC at point D, and DF being drawn perpendicular to BC and DG being drawn perpendicular to AB, where F and G represent the feet of the perpendiculars, if BG measures 4 units and AC measures 5 units, what is the perimeter of triangle ABC?	13	geoqa
<image>In the given diagram depicting right triangle ABC, where angle C measures 90 degrees, AC is 8.0, and AB is 10.0, point O lies on line segment BC and represents the center of a circle with radius OC measuring 1.0. As point D moves along the edge of line segment AB, a tangent DE is drawn to circle O at point E. What is the minimum length of line segment DE?	√{15}	geoqa
<image>R P = 3.5 and J P = 9. Find P S.	4.5	geoqa
<image>In the given figure, AB is a tangent to circle O and B is the point of tangency. If BC is the diameter of circle O and angle A equals 20.0, determine the measure of angle ODC. Denote the measure of angle ODC as α.	55°	geoqa
<image>Xiao Liang designed colorful flags to promote the Shanghai World Expo in 2010, with the given figure showing angle ACB measuring 90 degrees, angle D measuring 15 degrees, point A lying on CD, and AD equal to AB at 4.0 units. What is the length of AC?	2√{3}cπ	geoqa
<image>In the given diagram, if CE bisects angle BCD and is perpendicular to BD at point E, and if angle DAB equals angle ABD, with AC measuring 24.0 and the perimeter of triangle BCD being 34.0, what is the length of BD?	14	geoqa
<image>Find x.	8	geoqa
<image>In the diagram, Xiaoya's house (point O) is situated on an east-west road, and a water tower (point A) is positioned at 60.0 degrees 500.0 east of the north of her house. What is the distance from the water tower to the highway AB? Express the distance as 'd'.	250m	geoqa
<image>As shown in the diagram, in ABCD, the angle bisector of angle ABC intersects AD at point E. If AE = 2ED = 3, then the perimeter of ABCD is ().	15	geoqa
<image>What is the area of quadrilateral ABCD if AB measures 15 units, BC measures 12 units, CD measures 16 units, AD measures 25 units, and angle C is 90 degrees?	246	geoqa
<image>Triangle A B C is similar to triangle D E F. Find the scale factor.	\frac{2}{1}	geoqa
<image>In the given diagram, quadrilateral ABCD is inscribed in circle O. The diameter BE is part of circle O and AE is connected to it. If the measure of angle BCD is twice the measure of angle BAD, then what is the degree measure of angle DAE?	30°	geoqa
<image>In the given figure, the diagonals AC and BD of quadrilateral ABCD intersect at point O. If angle BAC measures 90 degrees, AC measures 6 units, and BD measures 10 units, what is the length of CD?	4	geoqa
<image>In the given diagram, it is depicted that circle O has diameter AB, where point C is the midpoint of the arc, and point D lies on the circle O on the opposite side of AB. What is the measurement of angle D?	45°	geoqa
<image>In the provided figure, a right-angled triangle board is positioned in such a way that angle F, angle ACB, and angle D all measure 90 degrees. Angle B is measured as 60 degrees and angle D is 45 degrees. Additionally, AB is parallel to DC. What is the degree measure of angle CAE?	15°	geoqa
<image>In the given diagram of a diamond-shaped ABCD, where angle BAD measures 110.0 degrees, the perpendicular bisector of AB intersects the diagonal AC at point F. The foot of the perpendicular is labeled E, and it is connected to DF. What is the measure of angle CDF?	15°	geoqa

<image>As shown in the picture, Xiaobin used a 50m long rope to enclose a parallelogram field. If one side is 16m long, then its adjacent side is ().

9m

geoqa

<image>Find m \angle 2.

24

geoqa

<image>In the provided diagram, parallelogram ABCD is shown where AB measures 5.0. The diagonals intersect at point O. Given that the perimeter of triangle OCD is 23.0, what is the sum of the lengths of the two diagonals of parallelogram ABCD?

36

geoqa

<image>If points C and D are situated on circle O with AB serving as the diameter, and angle ACD is measured at 15.0 degrees, what is the degree measure of angle BAD?

75°

geoqa

<image>Use parallelogram A B C D to find m \angle A F B.

131

geoqa

<image>Circle A has diameters D F and P G. If P A = 7, find P G.

14

geoqa

<image>In the provided figure, circle O has points A, B, and C. If angle AOB measures 70.0 degrees, what is the degree measure of angle ACB?

35°

geoqa

<image>In the provided figure, points A, B, C, and D lie on circle O, with BC being the diameter of circle O. If the measure of angle D is 36.0 degrees, what is the degree measure of angle BCA?

54°

geoqa

<image>What is the measure of angle 1 in triangle ABC, where angle C is 90 degrees, EF is parallel to AB, and angle B measures 39 degrees?

51°

geoqa

<image>In the given figure showing parallelogram ABCD, point P lies on side AD such that the ratio of AP to DP is 1.0:2.0. If the shaded region in the figure has an area of 3.0, what is the area of parallelogram ABCD?

9

geoqa

<image>Find the area of the figure. Round to the nearest tenth if necessary.

374.5

geoqa

<image>As shown in the figure, in the parallelogram ABCD, AB = 4.0, the bisector of angle BAD and the extended line of BC intersect at point E, and DC at point F, and DF: CF = 3.0, DG perpendicular AE, and the vertical foot is G , If DG = 1.0, then the length of AE is ()

\frac{16√{2}}{3}

geoqa

<image>In the provided diagram, points A, B, and E are all in a straight line. The square ABCD has a side length of 6 units, while the square BEFG has a side length of 8 units. Point H is the midpoint of the line segment DF. What is the length of line segment BH?

5√{2}

geoqa

<image>Find the area of the figure. Round to the nearest tenth if necessary.

156

geoqa

<image>In the provided figure of quadrilateral ABCD, where AB measures 4 units, BC measures 6 units, and AE measures 2 units, what is the length of AF, the altitude on side DC?

3

geoqa

<image>What is the measure of angle A if angle B is 60.0 degrees, angle AED is 45.0 degrees, and DE is parallel to BC in the given figure?

75°

geoqa

<image>In the given figure, line a is perpendicular to line c and line b is also perpendicular to line c. If angle 1 measures 70.0 degrees, what is the measure of angle 2?

70°

geoqa

<image>What is the measure of angle ACB if angles APB and APB both touch circle O and have a measure of 50.0 degrees, and point C lies on circle O?

65°

geoqa

<image>In the given figure, line AB is parallel to line CD. Additionally, the bisector EF of angle BEG intersects line CD at point F. If angle 1 measures 42.0, what is the measure of angle 2?

159°

geoqa

<image>As shown in the figure, in triangle ABC, point M is the midpoint of BC, AD is the bisector of exterior angle of triangle ABC, and AD perpendicular BD, if AB = 6.0, AC = 9.0, the length of MD is ()

\frac{15}{2}

geoqa

<image>In the given figure, circle O is depicted with chords AB and CD intersecting at point P. If the measure of angle A is 42.0 degrees and angle APD is 77.0 degrees, what is the measure of angle B?

35°

geoqa

<image>Consider the parallelogram ABCD with angle B measuring 110.0 degrees. If AD is extended to point F and CD is extended to point E, resulting in the connection of EF, what is the sum of angles E and F? Denote the sum as 'θ'.

70°

geoqa

<image>A B C D is a trapezoid. If A C = 3 x - 7 and B D = 2 x + 8, find the value of x so that A B C D is isosceles.

15

geoqa

<image>In the given diagram of triangle ABC, where EF is parallel to BC, and the ratio of AE to BE is 1:2, with EF having a length of 3.0 units, what is the value of BC?

9

geoqa

<image>What is the measure of angle BCF if, in the provided figure, AD is equal to BC and AE is equal to CF, E and F are points on BD, and BE is equal to DF, with angle AEB measuring 100 degrees and angle ADB measuring 30 degrees?

70°

geoqa

<image>In triangle ABC, points D and E lie on AB and AC respectively. DE is parallel to BC. If AD measures 5.0, BD measures 10.0, and AE measures 3.0, what is the length of AC?

9

geoqa

<image>In the given triangle ABC, with angle ACB measuring 90°, three squares are constructed outside the triangle using each side of the triangle. If the areas of these squares are represented by S1, S2, and S3, and S1 equals 3 while S2 equals 11, what is the value of S3?

8

geoqa

<image>Triangle J K L is circumscribed about \odot R. Find the perimeter of \triangle J K L.

52

geoqa

<image>P Q is tangent to the circle. Find x. Round to the nearest tenth.

5.2

geoqa

<image>In the given triangle ABC, points D and E are situated on the sides AC and BC respectively, with DE being parallel to AB. If the ratio of CD to CA is 2:3 and the area of triangle ABC is 18.0, what is the area of triangle DEC?

8

geoqa

<image>In the given triangle ABC, the midpoints of BC and AC are denoted by D and E, respectively. The angle ABC is bisected by BF, which intersects DE at point F. If the length of BC is 6 units, what is the length of DF?

3

geoqa

<image>In the provided figure, the ray OA has its endpoint O lying on the line CD. If angle COA is measured at 37.0 degrees, what is the measure of angle AOD?

143°

geoqa

<image>In the given diagram, circle O is depicted and AB is the diameter of circle O. The length of AB is equal to the square root of 2 times the length of AC. If angle ODC measures 12 degrees, what is the measure of angle DBC?

102°

geoqa

<image>In a garden, there is a pentagonal open space as depicted in the figure. To enhance the aesthetics, there is a plan to plant flowers and plants at each vertex of the pentagon, with each plant placed at the center of a circle with a radius of 2.0 units. What is the total area of the sector formed by each circle? Represent the total area as 'A'.

6πm²

geoqa

<image>In the given diagram, triangle ABC is congruent to triangle ADE. If the measure of angle BAE is 135° and the measure of angle DAC is 55°, what is the measure of angle CFE?

40°

geoqa

<image>In the provided figure, circle O has diameter AB, and chord CD is perpendicular to AB. Point E is the foot of the perpendicular. Point G can be anywhere on arc AC. By extending the line AG and DC to point F, and connecting GC, GD, and AD, what is the measure of angle AGD if angle BAD is 25.0 degrees?

65°

geoqa

<image>In the given diagram, if the line segment AB is the diameter of circle O and chord CD is perpendicular to AB, with angle CAB measuring 20.0, what is the measure of angle AOD?

140°

geoqa

<image>In the given figure, AB represents the diameter of circle O, while AC denotes the tangent of circle O, with point D where OC connects to circle O. If angle C is 36.0 degrees, what is the measure of angle B?

27°

geoqa

<image>In the given triangle ABC, with the angle bisector of angle ABC intersecting the perpendicular bisector of AC at point D, and DF being drawn perpendicular to BC and DG being drawn perpendicular to AB, where F and G represent the feet of the perpendiculars, if BG measures 4 units and AC measures 5 units, what is the perimeter of triangle ABC?

13

geoqa

<image>In the given diagram depicting right triangle ABC, where angle C measures 90 degrees, AC is 8.0, and AB is 10.0, point O lies on line segment BC and represents the center of a circle with radius OC measuring 1.0. As point D moves along the edge of line segment AB, a tangent DE is drawn to circle O at point E. What is the minimum length of line segment DE?

√{15}

geoqa

<image>R P = 3.5 and J P = 9. Find P S.

4.5

geoqa

<image>In the given figure, AB is a tangent to circle O and B is the point of tangency. If BC is the diameter of circle O and angle A equals 20.0, determine the measure of angle ODC. Denote the measure of angle ODC as α.

55°

geoqa

<image>Xiao Liang designed colorful flags to promote the Shanghai World Expo in 2010, with the given figure showing angle ACB measuring 90 degrees, angle D measuring 15 degrees, point A lying on CD, and AD equal to AB at 4.0 units. What is the length of AC?

2√{3}cπ

geoqa

<image>In the given diagram, if CE bisects angle BCD and is perpendicular to BD at point E, and if angle DAB equals angle ABD, with AC measuring 24.0 and the perimeter of triangle BCD being 34.0, what is the length of BD?

14

geoqa

<image>Find x.

8

geoqa

<image>In the diagram, Xiaoya's house (point O) is situated on an east-west road, and a water tower (point A) is positioned at 60.0 degrees 500.0 east of the north of her house. What is the distance from the water tower to the highway AB? Express the distance as 'd'.

250m

geoqa

<image>As shown in the diagram, in ABCD, the angle bisector of angle ABC intersects AD at point E. If AE = 2ED = 3, then the perimeter of ABCD is ().

15

geoqa

<image>What is the area of quadrilateral ABCD if AB measures 15 units, BC measures 12 units, CD measures 16 units, AD measures 25 units, and angle C is 90 degrees?

246

geoqa

<image>Triangle A B C is similar to triangle D E F. Find the scale factor.

\frac{2}{1}

geoqa

<image>In the given diagram, quadrilateral ABCD is inscribed in circle O. The diameter BE is part of circle O and AE is connected to it. If the measure of angle BCD is twice the measure of angle BAD, then what is the degree measure of angle DAE?

30°

geoqa

<image>In the given figure, the diagonals AC and BD of quadrilateral ABCD intersect at point O. If angle BAC measures 90 degrees, AC measures 6 units, and BD measures 10 units, what is the length of CD?

4

geoqa

<image>In the given diagram, it is depicted that circle O has diameter AB, where point C is the midpoint of the arc, and point D lies on the circle O on the opposite side of AB. What is the measurement of angle D?

45°

geoqa

<image>In the provided figure, a right-angled triangle board is positioned in such a way that angle F, angle ACB, and angle D all measure 90 degrees. Angle B is measured as 60 degrees and angle D is 45 degrees. Additionally, AB is parallel to DC. What is the degree measure of angle CAE?

15°

geoqa

<image>In the given diagram of a diamond-shaped ABCD, where angle BAD measures 110.0 degrees, the perpendicular bisector of AB intersects the diagonal AC at point F. The foot of the perpendicular is labeled E, and it is connected to DF. What is the measure of angle CDF?

15°

geoqa