Guizhen/geo170k_reasoning · Datasets at Hugging Face

images images listlengths 1 1	problem stringlengths 7 719	answer stringlengths 0 22
	<image>What is the value of tanθ if the length of an escalator in a supermarket is 13.0 and the maximum height reached by the escalator is 5.0?	\frac{5}{12}
	<image>For the pair of similar figures, use the given areas to find x.	35
	<image>In the given figure, a triangle ruler and its center projection under light form a similarity figure. If the scale factor of similarity is 1.0:2.0 and the length of one side of the triangle ruler is 5.0 units, what is the length of the corresponding edge of the projection triangle?	10cm
	<image>In the provided figure, circle O is shown with point P located outside the circle. PA is a tangent to circle O. Point C lies on circle O, and PC passes through the center O and intersects the circle at point B. If the measure of angle P is 46.0 degrees, what is the measure of angle ACP?	22°
	<image>In the given figure, the intersection of lines AC and BD is represented by point O in rectangle ABCD. If the area of triangle AOB is 4, what is the area of rectangle ABCD?	16
	<image>Find the area of the parallelogram. Round to the nearest tenth if necessary.	115
	<image>What is the degree measure of angle BDC if B is the midpoint of arc AC in circle O, D is a point on the circle, and angle AOB measures 50 degrees?	25°
	<image>In the provided figure, circle O is depicted, where OA is perpendicular to BC. If angle AOB measures 70 degrees, what is the degree measure of angle ADC?	35°
	<image>Polygon B C D E \cong polygon R S T U. Find z.	3
	<image>In the provided figure, PA and PB are tangent lines to circle O. If angle E measures 60.0 degrees, find the measure of angle P denoted as θ.	60°
	<image>In the provided figure, the perimeter of parallelogram ABCD is given to be 22.0 and the perimeter of triangle ABC is given to be 17.0. What is the length of AC?	6cm
	<image>As shown in the figure, in triangle ABC, points D and E are points on sides AB and AC respectively, DE bisects AB perpendicularly, angle C = 90.0, angle BAC = 15.0. If BC = 3.0, then the length of AE is ()	6cm
	<image>In the provided figure, triangle ABC is positioned within circle O, with OD perpendicular to BC at point D. If angle DOC measures 50.0 degrees, what is the degree measure of angle A?	50°
	<image>Find x.	6
	<image>Find x.	24\sqrt{2}
	<image>In the provided figure, the right-angled trapezoid ABCD is translated in the direction of AD to form the trapezoid EFGH. Given that HG measures 24.0 units, MG measures 8.0 units, and MC measures 6.0 units, what is the area of the shadowed region?	168
	<image>Find m \angle J L K.	40
	<image>In the given diagram, line a and line b are parallel, with the right-angle vertex of triangle board placed on line b. The two perpendicular sides intersect line a. If angle 2 measures 27°, what is the measure of angle 1?	63°
	<image>If points A, B, and C lie on circle O and angle AOB measures 70.0 degrees, what is the degree measure of angle ACB?	35°
	<image>Find m \angle 2.	23
	<image>In the given illustration, the diagonals of the diamond ABCD measure 2 and 5. If point P lies on diagonal AC (not the same as points A or C), and if PE is parallel to BC intersecting AB at E, and PF is parallel to CD intersecting AD at F, what is the area of the shaded region?	2.5
	<image>In the given figure, BC is the diameter of circle O, and the chord BD passes through point B parallel to the radius OA. If the measure of angle B is 50.0, then determine the measure of angle C.	25°
	<image>Find D C if D G and J M are altitudes and \triangle K J L \sim \triangle E D C.	4
	<image>As shown in the figure, in Rttriangle ABC, angle C = 90.0, BC = 6.0, AC = 8.0, then the value of sinA is equal to ()	\frac{3}{5}
	<image>Find m \angle W.	74
	<image>In the given figure, rectangle ABCD is depicted with intersecting lines AC and BD at point O. The midpoints of AB and OA are represented by M and N, respectively. If MN has a length of 2, what is the length of AC?	8
	<image>In a scenario where the height of a tree needs to be measured from ground level by the school mathematics interest group, what measurement scheme was devised using a mirror and a measuring tape as per the law of light reflection in "Science"?	5.6米
	<image>What is the measure of angle BAE in the rectangle ABCD shown in the figure, where diagonal AC intersects with BD at point O and a perpendicular line AE is drawn from point A to BD, with the foot of the perpendicular being point E, given that angle EAC is twice the measure of angle CAD?	22.5°
	<image>If angle AOB measures 120° and point C lies on the arc AB in the given diagram, what is the measure of angle ACB?	120°
	<image>What is the measure of angle AOE in the rhombus ABCD, where angle ABC is 50° and the diagonals AC and BD intersect at point O, with E being the midpoint of CD?	115°
	<image>In the given triangle ABC, if the measure of angle B is 40°, and the perpendicular bisector of side AC intersects AC at point D and BC at point E, with the angle EAB to CAE ratio being 3, what is the measure of angle C?	28°
	<image>In the given diagram, triangle ABC is depicted with point D serving as the midpoint of side BC and point E serving as the midpoint of side AD. If lines BE and CE are connected, and the area of triangle ABC is known to be 20, what is the area of triangle BCE?	10
	<image>What is the length of BE in parallelogram ABCD if AD measures 8.0 and AB measures 6.0, and if angle ADC is bisected by line segment DE?	2cm
	<image>If triangle ABC is folded along line segment DE and the sum of angle 1 and angle 2 is 60.0, what is the measure of angle A?	30°
	<image>Find the area of the quadrilateral.	499.5
	<image>What is the perimeter of triangle CDE in the given figure, where triangle ABC is depicted with AB = AC = 12, BC = 8, and AD bisects angle BAC, intersecting BC at point D, and E is the midpoint of AC?	16
	<image>As shown in the figure, draw the tangent AB of the circle from a point A outside circle O, connect AO and extend to intersect circle at point C, connect BC. Given that angle A = 26.0, then the degree of angle ACB is ()	32°
	<image>What is the length of BC in triangle ABC if vertices D and E are located on the sides AB and AC respectively, with the ratios AD/BD = AE/EC = 1/2 and DE measuring 3.0?	9
	<image>What is the length of BD if the vertices A, B, and C of the diamond-shaped OABC lie on the circle O, and the tangent line of circle O passes through point B and intersects the extension line of OA at point D, given that the radius of circle O is 1?	√{3}
	<image>In the provided diagram, where AB is parallel to CD and CD is parallel to EF, and with angle ABE measuring 70.0 degrees and angle DCE measuring 144.0 degrees, what is the measure of angle BEC?	34°
	<image>In the given diagram, triangle ABC is depicted with points D, E, and F marking the midpoints of sides BC, AD, and CE respectively. If the area of triangle ABC is measured at 12 square cm, what is the area of shaded triangle AEF?	1.5
	<image>Find y.	12
	<image>Given the diagram with angle 1 measuring 65°, angle B measuring 65°, and angle C measuring 80°, what is the degree measure of angle 2?	100°
	<image>Find \angle 7.	98
	<image>In the figure, in triangle ABC, the perpendicular bisector of side AB intersects side AC at point E and intersects side AB at point D. If the length of AC is 16cm and the length of BE is 12cm, then the length of EC is ().	4cm
	<image>In \odot Z, \angle W Z X \cong \angle X Z Y, m \angle V Z U = 4 x, m \angle U Z Y = 2 x + 24, and V Y and W U are diameters. Find m \widehat W X.	52
	<image>Find x so that the quadrilateral is a parallelogram.	4
	<image>In the provided diagram, point D lies on line segment BC and point E lies on line segment AC. AD and BE are connected. Given that angle A measures 35°, angle B measures 25°, and angle C measures 50°, what is the measure of angle 1?	70°
	<image>In the given figure, with AB being parallel to CD and AC intersecting BD at point O, if the lengths of DO, BO, and DC are known to be 2, 5, and 3 respectively, what is the length of AB?	7.5
	<image>In the provided figure, circle O has a diameter of 10 units and chord AB has a length of 6 units. As point M moves along the chord AB, what is the range of the length of line segment OM?	4≤OM≤5
	<image>Find x.	\frac{15\sqrt{2}}{2}
	<image>In the provided figure, triangle ABC has points D, E, and F as the midpoints of sides BC, AD, and CE respectively. The area of the shaded region S is determined to be 3cm2. What is the area of triangle ABC?	12
	<image>If AB is the diameter of circle O and points C, D, and E are located on circle O, with angle AED measuring 25°, what is the measurement of angle BCD?	115°
	<image>In the given diagram, it is given that AB is parallel to CD. The angle bisectors of angle ABE and angle CDE intersect at point F. If angle BFD measures 130°, what is the degree measure of angle BED?	100°
	<image>In the given diagram, triangle ABC has BD as its median. If AB measures 6 units and BC measures 4 units, what is the difference in perimeter between triangle ABD and triangle BCD?	2
	<image>Find x.	5.4
	<image>Given that DE is parallel to BC in triangle ABC and the ratio of AD to DB is 3.0 to 2.0, with a length of DE measuring 6.0 cm, what is the length of BC?	10cm
	<image>The segment is tangent to the circle. Find x.	3
	<image>In the given diagram, if DE is parallel to BC and BE bisects angle ABC, what is the degree measure of angle CBE if angle 1 has a measure of 66°?	33°
	<image>Find the area of the figure. Round to the nearest tenth if necessary.	500
	<image>Find x.	12
	<image>As shown in the diagram, in triangle ABC, points D and E are on the perpendicular bisectors of AB and AC respectively. If BC = 24, the perimeter of triangle ADE is ()	24
	<image>If x = 7 \sqrt { 3 }, find b.	42
	<image>X Y and X Z are midsegments of \triangle R S T. Find m \angle R Y X.	124
	<image>In the given diagram, if AB is parallel to CD and angle DCF measures 100.0 degrees, what is the measure of angle AEF?	80°
	<image>In the provided figure, two vertices of a right-angled triangle ruler, with a 30° angle, are positioned on opposite sides of a ruler. If angle 1 measures 20°, what is the degree measurement of angle 2?	40°
	<image>Find m \angle C.	103
	<image>In the provided diagram, triangle ABC is a right triangle with angle ACB measuring 90°. The line segment CD represents the altitude on side AB, and CE represents the median on side AB. Given that AD measures 2 units and CE measures 10 units, what is the length of CD?	6
	<image>What is the distance between the fishing boat and lighthouse C when, after sailing 12.0 nautical miles eastward from point A to point B, lighthouse C is seen facing up due to the north?	4√{3}海里
	<image>Use parallelogram A B C D to find m \angle A F B.	131
	<image>Find m \angle C.	127
	<image>In the given figure, M and F are points on the circle A. The perpendicular bisector of AM intersects circle A at points B and C, and the line segment AF intersects at point D, connecting DM. If angle MBF measures 20.0 degrees, what is the measure of angle DMF? Represent the measure of angle DMF as ϕ.	30°
	<image>, the shadow length of BD on the ground is 3.0, and the distance from D to the ground is 20.0?	\frac{80}{3}m
	<image>Find the area of the shaded sector. Round to the nearest tenth, if necessary.	40.2
	<image>As shown in the figure, AB is parallel to CD, CE bisects angle ACD, angle A is 108°. The measure of angle BEC is ()	144°
	<image>Find h.	6
	<image> In a pentagon ABCDE that is similar to pentagon POGMN, the ratio of the corresponding heights is 3.0:2.0. What is the area ratio of pentagon ABCDE to pentagon POGMN?	9:4
	<image>Find m \angle 1.	48
	<image>Find y.	3
	<image>Refer to trapezoid C D F G with median H E. Let Y Z be the median of H E F G. Find Y Z.	34
	<image>P is the centroid of triangle Q U S. If Q P = 14 centimeters, what is the length of Q T?	21
	<image>What is the length of the side of square ABCD if vertex C lies on the straight line a and vertices B and D are 1.0 and 2.0 units away from line a, respectively?	√{5}
	<image>Given the points A, B, and C on circle O, find the degree measure of angle BOC if angle BAC measures 30.0 degrees.	60°
	<image>In the provided diagram, consider a chord AB of circle O (with AB not being the diameter). Using point A as the center and AB as the radius, draw an arc that intersects circle O at point C. Next, connect AC, BC, OB, and OC. Given that angle ABC measures 65.0 degrees, what is the measure of angle BOC?	100°
	<image>Xiaogang uses a piece of sector cardboard with a radius of 24.0 to make the lateral surface of a conical clown hat as shown in the figure (the seams are ignored). If the bottom radius of the made cone-shaped clown hat is 10.0, then this sector cardboard is Area is ()	240πcm²
	<image>W X Y Z is a kite. If m \angle W X Y = 120, m \angle W Z Y = 4 x, and m \angle Z W X = 10 x, find m \angle Z Y X.	100
	<image>In the given diagram, if BC is the diameter of circle O and AD is tangent to circle O at point A, and angle C measures 40.0 degrees, what is the measure of angle DAC?	50°
	<image>As shown in the figure, the length of diagonal AC of rectangle ABCD is 8, and angle BOC is 120°. Therefore, the length of AB is ().	4
	<image>If b = 18, find c.	12\sqrt{3}
	<image>In right-angled triangle XYZ, where angle YZX = 90.0, the line PQ passing through point Z is parallel to the median MR of angle XYZ. If angle X = 50.0, what is the measure of angle ZPQ?	70°
	<image>In the given diagram, point P is located inside triangle ABC, and D, E, F are the symmetric points of P with respect to the sides AB, BC, and AC, respectively. If angle A in triangle ABC measures 70.0 degrees, angle B measures 60.0 degrees, and angle C measures 50.0 degrees, what is the sum of angles ADB, BEC, and CFA?	360°
	<image>What is the measure of angle BAC in rhombus ABCD if angle D is measured at 150 degrees?	15°
	<image>In the provided figure, where AB, AC, and AD are equal in length and angle CBD measures 44.0 degrees, what is the measure of angle CAD?	88°
	<image>Find the area of the parallelogram. Round to the nearest tenth if necessary.	57.9
	<image>Find the perimeter of \parallelogram A B C D.	28
	<image>In the provided figure, triangle ABC is depicted with edge AC intersecting circle O at points C and D and passing through the center O. AB is tangent to circle O at point B. If angle A measures 34.0 degrees, what is the measure of angle C?	28°
	<image>Given points A, B, and C lying on circle O, consider the quadrilateral ABCO. If angle AOC measures 140.0 degrees, what is the measure of angle B? Represent the measure of angle B as θ.	110°
	<image>Triangle C D E has an area of 336 square inches. Find C E.	28
	<image>What is the degree measure of angle CDE if AB is parallel to DE, GF is perpendicular to BC at point F, and angle FGA measures 140°?	50°
	<image>In the provided figure, circle O is depicted with chord AC parallel to radius OB. If angle BOC measures 48.0 degrees, what is the degree measure of angle OAB?	24°

<image>What is the value of tanθ if the length of an escalator in a supermarket is 13.0 and the maximum height reached by the escalator is 5.0?

\frac{5}{12}

<image>For the pair of similar figures, use the given areas to find x.

35

<image>In the given figure, a triangle ruler and its center projection under light form a similarity figure. If the scale factor of similarity is 1.0:2.0 and the length of one side of the triangle ruler is 5.0 units, what is the length of the corresponding edge of the projection triangle?

10cm

<image>In the provided figure, circle O is shown with point P located outside the circle. PA is a tangent to circle O. Point C lies on circle O, and PC passes through the center O and intersects the circle at point B. If the measure of angle P is 46.0 degrees, what is the measure of angle ACP?

22°

<image>In the given figure, the intersection of lines AC and BD is represented by point O in rectangle ABCD. If the area of triangle AOB is 4, what is the area of rectangle ABCD?

16

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

115

<image>What is the degree measure of angle BDC if B is the midpoint of arc AC in circle O, D is a point on the circle, and angle AOB measures 50 degrees?

25°

<image>In the provided figure, circle O is depicted, where OA is perpendicular to BC. If angle AOB measures 70 degrees, what is the degree measure of angle ADC?

35°

<image>Polygon B C D E \cong polygon R S T U. Find z.

3

<image>In the provided figure, PA and PB are tangent lines to circle O. If angle E measures 60.0 degrees, find the measure of angle P denoted as θ.

60°

<image>In the provided figure, the perimeter of parallelogram ABCD is given to be 22.0 and the perimeter of triangle ABC is given to be 17.0. What is the length of AC?

6cm

<image>As shown in the figure, in triangle ABC, points D and E are points on sides AB and AC respectively, DE bisects AB perpendicularly, angle C = 90.0, angle BAC = 15.0. If BC = 3.0, then the length of AE is ()

6cm

<image>In the provided figure, triangle ABC is positioned within circle O, with OD perpendicular to BC at point D. If angle DOC measures 50.0 degrees, what is the degree measure of angle A?

50°

<image>Find x.

6

<image>Find x.

24\sqrt{2}

<image>In the provided figure, the right-angled trapezoid ABCD is translated in the direction of AD to form the trapezoid EFGH. Given that HG measures 24.0 units, MG measures 8.0 units, and MC measures 6.0 units, what is the area of the shadowed region?

168

<image>Find m \angle J L K.

40

<image>In the given diagram, line a and line b are parallel, with the right-angle vertex of triangle board placed on line b. The two perpendicular sides intersect line a. If angle 2 measures 27°, what is the measure of angle 1?

63°

<image>If points A, B, and C lie on circle O and angle AOB measures 70.0 degrees, what is the degree measure of angle ACB?

35°

<image>Find m \angle 2.

23

<image>In the given illustration, the diagonals of the diamond ABCD measure 2 and 5. If point P lies on diagonal AC (not the same as points A or C), and if PE is parallel to BC intersecting AB at E, and PF is parallel to CD intersecting AD at F, what is the area of the shaded region?

2.5

<image>In the given figure, BC is the diameter of circle O, and the chord BD passes through point B parallel to the radius OA. If the measure of angle B is 50.0, then determine the measure of angle C.

25°

<image>Find D C if D G and J M are altitudes and \triangle K J L \sim \triangle E D C.

4

<image>As shown in the figure, in Rttriangle ABC, angle C = 90.0, BC = 6.0, AC = 8.0, then the value of sinA is equal to ()

\frac{3}{5}

<image>Find m \angle W.

74

<image>In the given figure, rectangle ABCD is depicted with intersecting lines AC and BD at point O. The midpoints of AB and OA are represented by M and N, respectively. If MN has a length of 2, what is the length of AC?

8

<image>In a scenario where the height of a tree needs to be measured from ground level by the school mathematics interest group, what measurement scheme was devised using a mirror and a measuring tape as per the law of light reflection in "Science"?

5.6米

<image>What is the measure of angle BAE in the rectangle ABCD shown in the figure, where diagonal AC intersects with BD at point O and a perpendicular line AE is drawn from point A to BD, with the foot of the perpendicular being point E, given that angle EAC is twice the measure of angle CAD?

22.5°

<image>If angle AOB measures 120° and point C lies on the arc AB in the given diagram, what is the measure of angle ACB?

120°

<image>What is the measure of angle AOE in the rhombus ABCD, where angle ABC is 50° and the diagonals AC and BD intersect at point O, with E being the midpoint of CD?

115°

<image>In the given triangle ABC, if the measure of angle B is 40°, and the perpendicular bisector of side AC intersects AC at point D and BC at point E, with the angle EAB to CAE ratio being 3, what is the measure of angle C?

28°

<image>In the given diagram, triangle ABC is depicted with point D serving as the midpoint of side BC and point E serving as the midpoint of side AD. If lines BE and CE are connected, and the area of triangle ABC is known to be 20, what is the area of triangle BCE?

10

<image>What is the length of BE in parallelogram ABCD if AD measures 8.0 and AB measures 6.0, and if angle ADC is bisected by line segment DE?

2cm

<image>If triangle ABC is folded along line segment DE and the sum of angle 1 and angle 2 is 60.0, what is the measure of angle A?

30°

<image>Find the area of the quadrilateral.

499.5

<image>What is the perimeter of triangle CDE in the given figure, where triangle ABC is depicted with AB = AC = 12, BC = 8, and AD bisects angle BAC, intersecting BC at point D, and E is the midpoint of AC?

16

<image>As shown in the figure, draw the tangent AB of the circle from a point A outside circle O, connect AO and extend to intersect circle at point C, connect BC. Given that angle A = 26.0, then the degree of angle ACB is ()

32°

<image>What is the length of BC in triangle ABC if vertices D and E are located on the sides AB and AC respectively, with the ratios AD/BD = AE/EC = 1/2 and DE measuring 3.0?

9

<image>What is the length of BD if the vertices A, B, and C of the diamond-shaped OABC lie on the circle O, and the tangent line of circle O passes through point B and intersects the extension line of OA at point D, given that the radius of circle O is 1?

√{3}

<image>In the provided diagram, where AB is parallel to CD and CD is parallel to EF, and with angle ABE measuring 70.0 degrees and angle DCE measuring 144.0 degrees, what is the measure of angle BEC?

34°

<image>In the given diagram, triangle ABC is depicted with points D, E, and F marking the midpoints of sides BC, AD, and CE respectively. If the area of triangle ABC is measured at 12 square cm, what is the area of shaded triangle AEF?

1.5

<image>Find y.

12

<image>Given the diagram with angle 1 measuring 65°, angle B measuring 65°, and angle C measuring 80°, what is the degree measure of angle 2?

100°

<image>Find \angle 7.

98

<image>In the figure, in triangle ABC, the perpendicular bisector of side AB intersects side AC at point E and intersects side AB at point D. If the length of AC is 16cm and the length of BE is 12cm, then the length of EC is ().

4cm

<image>In \odot Z, \angle W Z X \cong \angle X Z Y, m \angle V Z U = 4 x, m \angle U Z Y = 2 x + 24, and V Y and W U are diameters. Find m \widehat W X.

52

<image>Find x so that the quadrilateral is a parallelogram.

4

<image>In the provided diagram, point D lies on line segment BC and point E lies on line segment AC. AD and BE are connected. Given that angle A measures 35°, angle B measures 25°, and angle C measures 50°, what is the measure of angle 1?

70°

<image>In the given figure, with AB being parallel to CD and AC intersecting BD at point O, if the lengths of DO, BO, and DC are known to be 2, 5, and 3 respectively, what is the length of AB?

7.5

<image>In the provided figure, circle O has a diameter of 10 units and chord AB has a length of 6 units. As point M moves along the chord AB, what is the range of the length of line segment OM?

4≤OM≤5

<image>Find x.

\frac{15\sqrt{2}}{2}

<image>In the provided figure, triangle ABC has points D, E, and F as the midpoints of sides BC, AD, and CE respectively. The area of the shaded region S is determined to be 3cm2. What is the area of triangle ABC?

12

<image>If AB is the diameter of circle O and points C, D, and E are located on circle O, with angle AED measuring 25°, what is the measurement of angle BCD?

115°

<image>In the given diagram, it is given that AB is parallel to CD. The angle bisectors of angle ABE and angle CDE intersect at point F. If angle BFD measures 130°, what is the degree measure of angle BED?

100°

<image>In the given diagram, triangle ABC has BD as its median. If AB measures 6 units and BC measures 4 units, what is the difference in perimeter between triangle ABD and triangle BCD?

2

<image>Find x.

5.4

<image>Given that DE is parallel to BC in triangle ABC and the ratio of AD to DB is 3.0 to 2.0, with a length of DE measuring 6.0 cm, what is the length of BC?

10cm

<image>The segment is tangent to the circle. Find x.

3

<image>In the given diagram, if DE is parallel to BC and BE bisects angle ABC, what is the degree measure of angle CBE if angle 1 has a measure of 66°?

33°

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

500

<image>Find x.

12

<image>As shown in the diagram, in triangle ABC, points D and E are on the perpendicular bisectors of AB and AC respectively. If BC = 24, the perimeter of triangle ADE is ()

24

<image>If x = 7 \sqrt { 3 }, find b.

42

<image>X Y and X Z are midsegments of \triangle R S T. Find m \angle R Y X.

124

<image>In the given diagram, if AB is parallel to CD and angle DCF measures 100.0 degrees, what is the measure of angle AEF?

80°

<image>In the provided figure, two vertices of a right-angled triangle ruler, with a 30° angle, are positioned on opposite sides of a ruler. If angle 1 measures 20°, what is the degree measurement of angle 2?

40°

<image>Find m \angle C.

103

<image>In the provided diagram, triangle ABC is a right triangle with angle ACB measuring 90°. The line segment CD represents the altitude on side AB, and CE represents the median on side AB. Given that AD measures 2 units and CE measures 10 units, what is the length of CD?

6

<image>What is the distance between the fishing boat and lighthouse C when, after sailing 12.0 nautical miles eastward from point A to point B, lighthouse C is seen facing up due to the north?

4√{3}海里

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

131

<image>Find m \angle C.

127

<image>In the given figure, M and F are points on the circle A. The perpendicular bisector of AM intersects circle A at points B and C, and the line segment AF intersects at point D, connecting DM. If angle MBF measures 20.0 degrees, what is the measure of angle DMF? Represent the measure of angle DMF as ϕ.

30°

<image>, the shadow length of BD on the ground is 3.0, and the distance from D to the ground is 20.0?

\frac{80}{3}m

<image>Find the area of the shaded sector. Round to the nearest tenth, if necessary.

40.2

<image>As shown in the figure, AB is parallel to CD, CE bisects angle ACD, angle A is 108°. The measure of angle BEC is ()

144°

<image>Find h.

6

<image> In a pentagon ABCDE that is similar to pentagon POGMN, the ratio of the corresponding heights is 3.0:2.0. What is the area ratio of pentagon ABCDE to pentagon POGMN?

9:4

<image>Find m \angle 1.

48

<image>Find y.

3

<image>Refer to trapezoid C D F G with median H E. Let Y Z be the median of H E F G. Find Y Z.

34

<image>P is the centroid of triangle Q U S. If Q P = 14 centimeters, what is the length of Q T?

21

<image>What is the length of the side of square ABCD if vertex C lies on the straight line a and vertices B and D are 1.0 and 2.0 units away from line a, respectively?

√{5}

<image>Given the points A, B, and C on circle O, find the degree measure of angle BOC if angle BAC measures 30.0 degrees.

60°

<image>In the provided diagram, consider a chord AB of circle O (with AB not being the diameter). Using point A as the center and AB as the radius, draw an arc that intersects circle O at point C. Next, connect AC, BC, OB, and OC. Given that angle ABC measures 65.0 degrees, what is the measure of angle BOC?

100°

<image>Xiaogang uses a piece of sector cardboard with a radius of 24.0 to make the lateral surface of a conical clown hat as shown in the figure (the seams are ignored). If the bottom radius of the made cone-shaped clown hat is 10.0, then this sector cardboard is Area is ()

240πcm²

<image>W X Y Z is a kite. If m \angle W X Y = 120, m \angle W Z Y = 4 x, and m \angle Z W X = 10 x, find m \angle Z Y X.

100

<image>In the given diagram, if BC is the diameter of circle O and AD is tangent to circle O at point A, and angle C measures 40.0 degrees, what is the measure of angle DAC?

50°

<image>As shown in the figure, the length of diagonal AC of rectangle ABCD is 8, and angle BOC is 120°. Therefore, the length of AB is ().

4

<image>If b = 18, find c.

12\sqrt{3}

<image>In right-angled triangle XYZ, where angle YZX = 90.0, the line PQ passing through point Z is parallel to the median MR of angle XYZ. If angle X = 50.0, what is the measure of angle ZPQ?

70°

<image>In the given diagram, point P is located inside triangle ABC, and D, E, F are the symmetric points of P with respect to the sides AB, BC, and AC, respectively. If angle A in triangle ABC measures 70.0 degrees, angle B measures 60.0 degrees, and angle C measures 50.0 degrees, what is the sum of angles ADB, BEC, and CFA?

360°

<image>What is the measure of angle BAC in rhombus ABCD if angle D is measured at 150 degrees?

15°

<image>In the provided figure, where AB, AC, and AD are equal in length and angle CBD measures 44.0 degrees, what is the measure of angle CAD?

88°

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

57.9

<image>Find the perimeter of \parallelogram A B C D.

28

<image>In the provided figure, triangle ABC is depicted with edge AC intersecting circle O at points C and D and passing through the center O. AB is tangent to circle O at point B. If angle A measures 34.0 degrees, what is the measure of angle C?

28°

<image>Given points A, B, and C lying on circle O, consider the quadrilateral ABCO. If angle AOC measures 140.0 degrees, what is the measure of angle B? Represent the measure of angle B as θ.

110°

<image>Triangle C D E has an area of 336 square inches. Find C E.

28

<image>What is the degree measure of angle CDE if AB is parallel to DE, GF is perpendicular to BC at point F, and angle FGA measures 140°?

50°

<image>In the provided figure, circle O is depicted with chord AC parallel to radius OB. If angle BOC measures 48.0 degrees, what is the degree measure of angle OAB?

24°