Sicong/RL_toy_math_3 · Datasets at Hugging Face

images images listlengths 1 1	problem stringlengths 7 3.42k	answer stringlengths 1 40
	<image>Find the degree of ∠1 in the following diagrams; In Diagram A, ∠1 = __; in Diagram B, ∠1 = __; in Diagram C, ∠1 = __.	60°
	<image>ABCD conforms a square. DCEF is geometrically a square. How would you determine the perimeter of DFG?	11*√(2) + 22
	<image>DEFG conforms a square. What is the measurement of the perimeter of DEFG?	8*√(3)/3
	<image>ABCD forms a square. DCEF conforms a square. GHI is in the shape of an equilateral triangle. What would the perimeter of DFGIH be?	257
	<image>ABCD is identified as a parallelogram. ECF is an isosceles triangle where the sides EC and CF are identical in length. FCG is identified as a sector. Could you specify the length of arc FG in sector FCG?	√(3)*π/9
	<image>Side CD extending outward into an equilateral triangle from the rectangle. DEF is in the form of a sector. GEHI takes the form of a parallelogram. How do you find the perimeter of GEHI?	254
	<image>As shown in the image, the circumference of the circle is 125.6 centimeters. What is the perimeter of the square?	160
	<image>As shown in the image, Island A is to the north-east at an angle of 30° from Island B, Island C is to the north-east at an angle of 80° from Island B, and Island A is to the north-west at an angle of 40° from Island C. The angle ∠BAC from Island A looking towards Islands B and C is ________.	70°
	<image>CDE forms an equilateral triangle. DEFG conforms a parallelogram. What is the total length of the perimeter of HFI?	√(2) + 2
	<image>Side CD shapes into an equilateral triangle. DEF is defined as a sector. GEHI is a square. How long is HE in shape GEHI?	4*√(3)/3
	<image>ABCD is geometrically a parallelogram. DCEF is a square. DFG forms a sector. Arc HI creates a semi-circle. What is the total perimeter measurement of GFHI?	6*π + 24
	<image>ABCD is a parallelogram. DEF is an isosceles triangle with DE having the same length as EF. What is the measurement of FE in the shape DEF?	24
	<image>ACDE is a square. AEF is a sector. In sector AEF, what is the length of arc AF?	16√(3)π/3
	<image> As shown in the figure, △ABC ≌ △DCB. If ∠A = 80° and ∠DBC = 40°, then the measure of ∠DCA is ______ degrees.	20
	<image>ABC is defined as a sector. CBD is identified as a sector. DBEF is geometrically a parallelogram. Square FEGH contains a circle that is inscribed. How would you determine the perimeter of inscribed circle?	7*π
	<image>CDE makes up an equilateral triangle. GFHI is a parallelogram. IHJK is a parallelogram. What is the overall area of the shape IHJK?	42*√(2)
	<image>Side EF turns into an equilateral triangle. Side IJ is extended into an equilateral triangle shape inside the rectangle. How would you determine the perimeter of HGIKJ?	150 + 100*√(3)
	<image>Find the area of the shaded region.	4.28
	<image>ABCD takes the form of a parallelogram. ECF is an isosceles triangle where EC is the same length as CF. Side GH shapes into an equilateral triangle inside the rectangle. How much area does the shape FCGIH cover?	-8281√(3)/4 + 5915√(2)
	<image>As shown in the figure, △ABC is an equilateral triangle, and the curve CDEF is called the involute of the equilateral triangle. The centers of the arcs CD, DE, and EF are A, B, and C respectively. If AB = 1, then the length of the curve CDEF is (___).	4\pi
	<image>The following shows the route map of Dongdong from home to the library. (1) According to the route map, complete the table. \frac{}(2) What is Dongdong's average speed for the entire journey?	50 meters/minute
	<image>ABC is geometrically a sector. Side DE expanding into an equilateral triangle beyond the rectangle. GHI results in an equilateral triangle. What is the entire perimeter length of EFGIH?	198
	<image>Please provide the length of GF in shape DEFG.	6*√(2)
	<image>DCE is identified as a sector. Side FG extending into an equilateral triangle shape beyond the rectangle. Side IJ growing into a semi-circle outside the rectangle. How would you calculate the total area of the shape GHIJ?	9π/8 + 9√(3)
	<image> As shown in the figure, $$AC$$ and $$BD$$ intersect at $$P$$, $$AD$$ and $$BC$$ are extended to meet at point $$E$$, $$∠AEC = 37^{\circ}$$ and $$∠CAE = 31^{\circ}$$. Then the degree of $$∠APB$$ is ______.	99°
	<image>As shown in the figure, AB and CD intersect at point O. ∠1 = ∠2, and ∠3 : ∠1 = 8 : 1. Find the degree measure of ∠4.	36°
	<image>ABCD is a square. Side EF evolving into an equilateral triangle. Side HI stretches into an equilateral triangle within the rectangle. What would the perimeter of GEHJI be?	7
	<image> Circle O has two diameters AB⊥CD, ∠AOE=50°, and ∠DOF is 2 times the ∠BOF. (1) Find the measure of the central angle ∠EOF. (2) What is the ratio of the area of sector COF to the area of sector COE?	160°
	<image>ABCD is identified as a parallelogram. Side EF extends into an equilateral triangle outside the rectangle. FGHI is a parallelogram. Please provide the length of IH in shape FGHI.	84
	<image>As shown in the figure, point F is on the extension of side BC of triangle △ABC, and FD is perpendicular to AB at point D. Given that ∠A = 30° and ∠F = 40°, find the measure of ∠ACB.	100
	<image>Side EF extending into an equilateral triangle shape beyond the rectangle. Side HI stretches into an equilateral triangle within the rectangle. What is the total area measurement of the shape GEHJI?	4200 - 900*√(3)
	<image>CBD is defined as a sector. DBEF is geometrically a square. What is the perimeter of DBEF?	96
	<image>ABC is defined as a sector. EDF is an isosceles triangle where ED and DF are of equal length. Arc GH is a semi-circle. What is the arc length GH in the shape FDGH?	22*π
	<image>ABC is identified as a sector. CBDE conforms a square. DBF is a sector. Arc GH is a half-circle. What is the full area of the shape FBGH?	5544 - 1089*π/2
	<image>Given a number line on a piece of paper (as shown in the figure), fold the paper. (1) If the point representing 1 overlaps with the point representing -1, then the point representing -7 overlaps with the point representing ______. (2) If the point representing -1 overlaps with the point representing 5, then the point representing 13 overlaps with the point representing ______.	7
	<image>ABCD forms a square. DCEF is a square. DFGH is a square. DHIJ takes the form of a parallelogram. Could you determine the area of the entire shape DHIJ?	9*√(2)/2
	<image>The diagram shows a rectangular piece of paper that has been folded. In the diagram, ∠1 = 50 degrees. Can you determine the measure of ∠2?	65
	<image>ABC is defined as a sector. CBD is in the form of a sector. FEG is a sector. Can you tell me the area of the whole shape FEG?	108*π
	<image>ABC is defined as a sector. CBDE is identified as a parallelogram. EDF is an isosceles triangle where ED is the same length as DF. Arc GH constitutes a semi-circle. What is the total perimeter measurement of FDGH?	48*π + 264
	<image>ABC is an isosceles triangle with AB and BC being equal in length. CBDE takes the form of a parallelogram. In the shape EDFG, what is the measurement of GF?	3
	<image>As shown in the figure, given that AB ∥ CD, ∠1 = 35°, and BD bisects ∠ADC, find the measure of ∠A.	110
	<image> In the figure, in $$\triangle ABC$$, $$AB = AC$$ and $$AD$$ is the median of $$\triangle ABC$$. $$E$$ is the midpoint of $$AC$$. Connect $$DE$$, and let $$DF \perp AB$$ at $$F$$. Prove that: $$(1) \angle B = \angle EDC$$; $$(2) \angle BDF = \angle ADE$$.	∠B=∠EDC
	<image> As shown in the figure, there are two clock faces. (1) At 3 o'clock, the angle between the hour hand and the minute hand is ____ degrees. (2) At 5 o'clock, the angle between the hour hand and the minute hand is ____ degrees.	90
	<image>ABC is an isosceles triangle, with AB equalling BC. EDF is an isosceles triangle where the sides ED and DF are identical in length. What is the area of the entire shape EDF?	9*√(3)
	<image>Side CD shapes into an equilateral triangle. FEGH is geometrically a square. Can you tell me the area of the whole shape FEGH?	2500
	<image>Given that AB∥CD, ∠A=45°, and ∠C=75° as shown in the figure, then ∠M=___▲____.	30°
	<image> As shown in the figure, △ABC and △DEF are both equilateral triangles. D and E are located on AB and BC respectively. Please find a triangle similar to △DBE and provide a proof.	△GAD
	<image>ABCD conforms a parallelogram. DEF takes on a sector shape. How do you find the area of the entire shape DEF?	π/9
	<image>ABCD is geometrically a parallelogram. DCE is a sector. Side FG evolving into an equilateral triangle. Can you calculate the perimeter of ECFHG?	456
	<image>ABCD is identified as a parallelogram. DCEF is identified as a parallelogram. FEG takes on a sector shape. GEH is identified as a sector. What would the perimeter of GEH be?	7*π/2 + 14
	<image>CBDE is identified as a parallelogram. EDF is an isosceles triangle where ED and DF are of equal length. Side GH curves into a semi-circle. In shape FDGH, what is the length of GH?	2*π
	<image> As shown in the figure, given △ABE, the perpendicular bisectors \( m_1 \) and \( m_2 \) on sides AB and AE respectively intersect BE at points C and D, and BC = CD = DE. Find the measure of ∠BAE.	120°
	<image>In the following definition represented by : if there exists a quadrilateral where the sum of squares of two adjacent sides is equal to the square of one of its diagonals, then such a quadrilateral is called a Pythagorean quadrilateral. These two adjacent sides are referred to as the Pythagorean sides of the quadrilateral. (1) List the names of two special quadrilaterals that you have learned which are Pythagorean quadrilaterals; (2) As shown in the figure, triangle ABC is rotated 60° clockwise around vertex B to obtain triangle DBE. Connect AD and BC, with ∠DCB=30°. Prove that DC^{2}+ BC^{2}=AC^{2}, indicating that the quadrilateral is a Pythagorean quadrilateral.	Pythagorean Quadrilateral
	<image>Point $A,B,C,D,$ and $E$ are equally spaced on a minor arc of a circle. Points $E,F,G,H,I$ and $A$ are equally spaced on a minor arc of a second circle with center $C$ as shown in the figure below. The angle $\angle ABD$ exceeds $\angle AHG$ by $12^\circ$. Find the degree measure of $\angle BAG$.	58
	<image>The square $A B C D$ has area 80. The points $E, F, G$ and $H$ are on the sides of the square and $\mathrm{AE}=\mathrm{BF}=\mathrm{CG}=\mathrm{DH}$. How big is the area of the grey part, if $\mathrm{AE}=3 \times \mathrm{EB}$?	25
	<image>ABCD conforms a parallelogram. FEGH forms a square. FHI is an isosceles triangle, with FH equalling HI. Can you specify the length of the base FI in the isosceles triangle FHI?	8*√(3)
	<image>EFG results in an equilateral triangle. Square GEHI has a circle inscribed. What is the total area of the circle?	π/4
	<image>In the figure shown, all quadrilaterals are squares, and all triangles are right-angled triangles. The hypotenuse of the largest right-angled triangle is 9 cm. The sum of the areas of the four shaded squares is ____ cm². ______	81
	<image>As shown in the figure, if \( l_1 \parallel l_2 \) and \( \angle 1 = 50^\circ \), then \( \angle 2 = \_\_\_^\circ \).	130
	<image>CDE describes an equilateral triangle. Side GH shapes into an equilateral triangle. IGJ is an isosceles triangle, with IG equalling GJ. Can you tell me the length of the base IJ in the isosceles triangle IGJ?	80√(6 - 3√(2))/3
	<image>Side EF takes the shape of an equilateral triangle. Square FGHI has an inscribed circle. How do you find the perimeter of inscribed circle?	65*π
	<image>Side DE extending into an equilateral triangle shape beyond the rectangle. Side GH developing into an equilateral triangle outside. What is the area of the entire shape HIJK?	540*√(2)
	<image>As shown in the figure, in rectangle \(ABCD\), point \(E\) is on side \(AB\). △\(EBC\) is folded along the line \(CE\), so that point \(B\) falls on point \(B′\) on side \(AD\). When the folded figure is unfolded, if the perimeter of △\(AB′E\) is 4 cm and the perimeter of △\(B′DC\) is 11 cm, then the length of \(B′D\) is _________ cm.	3.5
	<image> As shown in the figure, the side length of square ABCD is 3, and the side length of square AEFG is 4. Given that \( S_1 = S_2 \), \( S_3 = S_4 \), and \( S_5 = S_6 \), what is the area of square DEHK?	25
	<image>Side EF is shaped into an equilateral triangle. In shape FGHI, what is the length of IH?	7
	<image> As shown in the figure, AB ∥ CD, CD ∥ EF, ∠BCE = 90°. Find the degree measure of ∠E - ∠B.	90°
	<image>In triangle $ABC$, $AB = AC = 5$ and $BC = 6$. Let $O$ be the circumcenter of triangle $ABC$. Find the area of triangle $OBC$.	\(\frac{21}{8}\)
	<image> As shown in the figure, points A, O, and B are on the same straight line, and OC bisects ∠AOD. ∠AOC + ∠EOB = 90°. (1) Find the measure of ∠COE. (2) Determine the relationship between ∠DOE and ∠EOB, and provide reasoning.	90°
	<image>Side EF forming an outward equilateral triangle beyond the rectangle. Square GEHI contains an inscribed circle. Please provide the length of IH in shape GEHI.	21
	<image> On a lake surface as calm as a mirror, there stands a beautiful red lotus, which is $$1$$ meter above the water. A strong gust of wind blows, causing the red lotus to be swept to the side until the flower is level with the water surface. Given that the horizontal distance moved by the red lotus is $$2$$ meters, what is the depth of the water here?	1.5
	<image>The two triangles in the figure are both isosceles right triangles. Find the area of the shaded region in the figure.	24.5
	<image> Cut the following image into 4 small squares. Find the perimeter and side length of each small square.	2
	<image>ABC is an isosceles triangle with AB having the same length as BC. CBD is an isosceles triangle in which CB equals BD in length. DBEF is identified as a parallelogram. What is the overall area measurement of the shape DBEF?	1470
	<image>Side CD converting into an equilateral triangle outside. ECF is an isosceles triangle, with EC equalling CF. What is the full perimeter of FCGH?	8*√(2) + 14
	<image>ACDE forms a square. DCF is an isosceles triangle, with DC equalling CF. How much is the area of the entire shape FCG?	5400*√(3)
	<image>ACD is an isosceles triangle with AC and CD being equal in length. DCE is defined as a sector. Side FG continues into a semi-circle within the rectangle. Please provide the length of FG in shape ECFG.	√(3)*π/3
	<image>ABCD is geometrically a square. EBFG takes the form of a parallelogram. GFHI is geometrically a square. In shape GFHI, how long is IH?	3*√(3)
	<image>ABCD is a parallelogram. DCE is an isosceles triangle in which DC and CE are equal. How would you determine the perimeter of ECF?	22*√(3) + 66
	<image>DCEF is geometrically a square. DFGH is geometrically a square. How do you calculate the area of the shape DFGH?	100
	<image>DCEF is identified as a parallelogram. Side GH takes the shape of an equilateral triangle. What is the entire perimeter length of FEGIH?	440
	<image>As shown in the figure , the perimeter of this rectangle is ____ decimeters.	10
	<image>ABC is an isosceles triangle, with AB equalling BC. CBDE is a square. EDF is an isosceles triangle with ED and DF being equal in length. How do you find the area of the entire shape EDF?	18
	<image>ABCD is geometrically a parallelogram. DCEF is geometrically a square. DFGH is geometrically a square. What is the overall perimeter of DFGH?	52
	<image> As shown in the figure, points A, B, and C are on circle O, AC ∥ OB, and ∠BOC = 40°. Then ∠ABO = ____ degrees.	20
	<image>As shown in the figure, in the circle ⊙O, OA ∥ BC, and ∠ACB = 20°. Find ∠1 = ____.	60
	<image>ACD is an isosceles triangle where the lengths of AC and CD are equal. What is the full area of the shape DCE?	4704*√(3)
	<image>ABCD is geometrically a square. EFG is geometrically an equilateral triangle. HEIJ is a parallelogram. How would you determine the perimeter of HEIJ?	5
	<image>According to the diagram, a large rectangle is formed by 6 squares. Given that the smallest square has an area of 1 cm², what is the area of the large rectangle in cm²?	143
	<image>As shown in the figure, in trapezoid ABCD, AD∥BC, AB=a, DC=b. The perpendicular bisector of side DC, EF, intersects side BC at E, and E is the midpoint of BC. Additionally, DE∥AB. Find the perimeter of trapezoid ABCD.	4a+b
	<image>As shown in the image, a rectangular piece of paper has a width AB = 8 cm and a length BC = 10 cm. The paper is folded so that vertex D falls on point F on edge BC (the fold line is AE). Find the length of EC.	3
	<image>ABCD is geometrically a square. ADE is an isosceles triangle where AD = DE. EDF is an isosceles triangle with ED and DF being equal in length. I need to know the length of the base EF in the isosceles triangle EDF.	56
	<image>As shown in the , an isosceles right triangle $$ABC$$ is rotated clockwise around point $$C$$ on a horizontal table surface to position $$A'B'C$$, such that points $$A$$, $$C$$, and $$B'$$ are collinear. The measure of the angle of rotation is ______ .	135
	<image>EFG constitutes an equilateral triangle. How do you find the area of the entire shape ADEGF?	4704√(3) + 4592√(6)
	<image>As shown in the figure, fold one side of the rectangle $$ABCD$$ along $$AD$$ such that point $$D$$ falls on point $$F$$ on side $$BC$$. Given that the crease $$AE=5 \sqrt {5}cm$$ and $$\tan ∠EFC= \dfrac {3}{4}$$, what is the perimeter of the rectangle $$ABCD$$?	36
	<image>As shown in the image, it is known that AB∥CD, BF bisects ∠ABE, DF bisects ∠CDE, and ∠BFD = 140°. The measure of ∠BED is ______.	80
	<image>As shown in the figure, AB is the diameter of circle O, and points C and D are on circle O. If ∠C = 20°, then the measure of ∠ABD is (__)	70
	<image> When hosting a TV show, the host stands at the golden section point of the stage to appear the most natural and graceful. As shown in the figure, if the stage AB is 20 meters long and the host is currently standing at point A, how many meters should the host walk from point A to appear the most natural and graceful? (Result accurate to 0.1 meters)______.	7.6
	<image>Known from the : As shown in the figure, ∠BOC = 2∠AOB, OD bisects ∠AOC, and ∠BOD = 20°, find the degree of ∠AOB.	40
	<image>ABC is an isosceles triangle, having AB and BC of the same length. CBD is in the form of a sector. DBE is an isosceles triangle, with DB equalling BE. How long is the perimeter of EBFG?	38
	<image>Carefully observe the angles in the image.	1

<image>Find the degree of ∠1 in the following diagrams; In Diagram A, ∠1 = __; in Diagram B, ∠1 = __; in Diagram C, ∠1 = __.

60°

<image>ABCD conforms a square. DCEF is geometrically a square. How would you determine the perimeter of DFG?

11*√(2) + 22

<image>DEFG conforms a square. What is the measurement of the perimeter of DEFG?

8*√(3)/3

<image>ABCD forms a square. DCEF conforms a square. GHI is in the shape of an equilateral triangle. What would the perimeter of DFGIH be?

257

<image>ABCD is identified as a parallelogram. ECF is an isosceles triangle where the sides EC and CF are identical in length. FCG is identified as a sector. Could you specify the length of arc FG in sector FCG?

√(3)*π/9

<image>Side CD extending outward into an equilateral triangle from the rectangle. DEF is in the form of a sector. GEHI takes the form of a parallelogram. How do you find the perimeter of GEHI?

254

<image>As shown in the image, the circumference of the circle is 125.6 centimeters. What is the perimeter of the square?

160

<image>As shown in the image, Island A is to the north-east at an angle of 30° from Island B, Island C is to the north-east at an angle of 80° from Island B, and Island A is to the north-west at an angle of 40° from Island C. The angle ∠BAC from Island A looking towards Islands B and C is ________.

70°

<image>CDE forms an equilateral triangle. DEFG conforms a parallelogram. What is the total length of the perimeter of HFI?

√(2) + 2

<image>Side CD shapes into an equilateral triangle. DEF is defined as a sector. GEHI is a square. How long is HE in shape GEHI?

4*√(3)/3

<image>ABCD is geometrically a parallelogram. DCEF is a square. DFG forms a sector. Arc HI creates a semi-circle. What is the total perimeter measurement of GFHI?

6*π + 24

<image>ABCD is a parallelogram. DEF is an isosceles triangle with DE having the same length as EF. What is the measurement of FE in the shape DEF?

24

<image>ACDE is a square. AEF is a sector. In sector AEF, what is the length of arc AF?

16*√(3)*π/3

<image> As shown in the figure, △ABC ≌ △DCB. If ∠A = 80° and ∠DBC = 40°, then the measure of ∠DCA is ______ degrees.

20

<image>ABC is defined as a sector. CBD is identified as a sector. DBEF is geometrically a parallelogram. Square FEGH contains a circle that is inscribed. How would you determine the perimeter of inscribed circle?

7*π

<image>CDE makes up an equilateral triangle. GFHI is a parallelogram. IHJK is a parallelogram. What is the overall area of the shape IHJK?

42*√(2)

<image>Side EF turns into an equilateral triangle. Side IJ is extended into an equilateral triangle shape inside the rectangle. How would you determine the perimeter of HGIKJ?

150 + 100*√(3)

<image>Find the area of the shaded region.

4.28

<image>ABCD takes the form of a parallelogram. ECF is an isosceles triangle where EC is the same length as CF. Side GH shapes into an equilateral triangle inside the rectangle. How much area does the shape FCGIH cover?

-8281*√(3)/4 + 5915*√(2)

<image>As shown in the figure, △ABC is an equilateral triangle, and the curve CDEF is called the involute of the equilateral triangle. The centers of the arcs CD, DE, and EF are A, B, and C respectively. If AB = 1, then the length of the curve CDEF is (___).

4\pi

<image>The following shows the route map of Dongdong from home to the library. (1) According to the route map, complete the table. \frac{}(2) What is Dongdong's average speed for the entire journey?

50 meters/minute

<image>ABC is geometrically a sector. Side DE expanding into an equilateral triangle beyond the rectangle. GHI results in an equilateral triangle. What is the entire perimeter length of EFGIH?

198

<image>Please provide the length of GF in shape DEFG.

6*√(2)

<image>DCE is identified as a sector. Side FG extending into an equilateral triangle shape beyond the rectangle. Side IJ growing into a semi-circle outside the rectangle. How would you calculate the total area of the shape GHIJ?

9*π/8 + 9*√(3)

<image> As shown in the figure, $$AC$$ and $$BD$$ intersect at $$P$$, $$AD$$ and $$BC$$ are extended to meet at point $$E$$, $$∠AEC = 37^{\circ}$$ and $$∠CAE = 31^{\circ}$$. Then the degree of $$∠APB$$ is ______.

99°

<image>As shown in the figure, AB and CD intersect at point O. ∠1 = ∠2, and ∠3 : ∠1 = 8 : 1. Find the degree measure of ∠4.

36°

<image>ABCD is a square. Side EF evolving into an equilateral triangle. Side HI stretches into an equilateral triangle within the rectangle. What would the perimeter of GEHJI be?

7

<image> Circle O has two diameters AB⊥CD, ∠AOE=50°, and ∠DOF is 2 times the ∠BOF. (1) Find the measure of the central angle ∠EOF. (2) What is the ratio of the area of sector COF to the area of sector COE?

160°

<image>ABCD is identified as a parallelogram. Side EF extends into an equilateral triangle outside the rectangle. FGHI is a parallelogram. Please provide the length of IH in shape FGHI.

84

<image>As shown in the figure, point F is on the extension of side BC of triangle △ABC, and FD is perpendicular to AB at point D. Given that ∠A = 30° and ∠F = 40°, find the measure of ∠ACB.

100

<image>Side EF extending into an equilateral triangle shape beyond the rectangle. Side HI stretches into an equilateral triangle within the rectangle. What is the total area measurement of the shape GEHJI?

4200 - 900*√(3)

<image>CBD is defined as a sector. DBEF is geometrically a square. What is the perimeter of DBEF?

96

<image>ABC is defined as a sector. EDF is an isosceles triangle where ED and DF are of equal length. Arc GH is a semi-circle. What is the arc length GH in the shape FDGH?

22*π

<image>ABC is identified as a sector. CBDE conforms a square. DBF is a sector. Arc GH is a half-circle. What is the full area of the shape FBGH?

5544 - 1089*π/2

<image>Given a number line on a piece of paper (as shown in the figure), fold the paper. (1) If the point representing 1 overlaps with the point representing -1, then the point representing -7 overlaps with the point representing ______. (2) If the point representing -1 overlaps with the point representing 5, then the point representing 13 overlaps with the point representing ______.

7

<image>ABCD forms a square. DCEF is a square. DFGH is a square. DHIJ takes the form of a parallelogram. Could you determine the area of the entire shape DHIJ?

9*√(2)/2

<image>The diagram shows a rectangular piece of paper that has been folded. In the diagram, ∠1 = 50 degrees. Can you determine the measure of ∠2?

65

<image>ABC is defined as a sector. CBD is in the form of a sector. FEG is a sector. Can you tell me the area of the whole shape FEG?

108*π

<image>ABC is defined as a sector. CBDE is identified as a parallelogram. EDF is an isosceles triangle where ED is the same length as DF. Arc GH constitutes a semi-circle. What is the total perimeter measurement of FDGH?

48*π + 264

<image>ABC is an isosceles triangle with AB and BC being equal in length. CBDE takes the form of a parallelogram. In the shape EDFG, what is the measurement of GF?

3

<image>As shown in the figure, given that AB ∥ CD, ∠1 = 35°, and BD bisects ∠ADC, find the measure of ∠A.

110

<image> In the figure, in $$\triangle ABC$$, $$AB = AC$$ and $$AD$$ is the median of $$\triangle ABC$$. $$E$$ is the midpoint of $$AC$$. Connect $$DE$$, and let $$DF \perp AB$$ at $$F$$. Prove that: $$(1) \angle B = \angle EDC$$; $$(2) \angle BDF = \angle ADE$$.

∠B=∠EDC

<image> As shown in the figure, there are two clock faces. (1) At 3 o'clock, the angle between the hour hand and the minute hand is ____ degrees. (2) At 5 o'clock, the angle between the hour hand and the minute hand is ____ degrees.

90

<image>ABC is an isosceles triangle, with AB equalling BC. EDF is an isosceles triangle where the sides ED and DF are identical in length. What is the area of the entire shape EDF?

9*√(3)

<image>Side CD shapes into an equilateral triangle. FEGH is geometrically a square. Can you tell me the area of the whole shape FEGH?

2500

<image>Given that AB∥CD, ∠A=45°, and ∠C=75° as shown in the figure, then ∠M=___▲____.

30°

<image> As shown in the figure, △ABC and △DEF are both equilateral triangles. D and E are located on AB and BC respectively. Please find a triangle similar to △DBE and provide a proof.

△GAD

<image>ABCD conforms a parallelogram. DEF takes on a sector shape. How do you find the area of the entire shape DEF?

π/9

<image>ABCD is geometrically a parallelogram. DCE is a sector. Side FG evolving into an equilateral triangle. Can you calculate the perimeter of ECFHG?

456

<image>ABCD is identified as a parallelogram. DCEF is identified as a parallelogram. FEG takes on a sector shape. GEH is identified as a sector. What would the perimeter of GEH be?

7*π/2 + 14

<image>CBDE is identified as a parallelogram. EDF is an isosceles triangle where ED and DF are of equal length. Side GH curves into a semi-circle. In shape FDGH, what is the length of GH?

2*π

<image> As shown in the figure, given △ABE, the perpendicular bisectors $ m_1 $ and $ m_2 $ on sides AB and AE respectively intersect BE at points C and D, and BC = CD = DE. Find the measure of ∠BAE.

120°

<image>In the following definition represented by : if there exists a quadrilateral where the sum of squares of two adjacent sides is equal to the square of one of its diagonals, then such a quadrilateral is called a Pythagorean quadrilateral. These two adjacent sides are referred to as the Pythagorean sides of the quadrilateral. (1) List the names of two special quadrilaterals that you have learned which are Pythagorean quadrilaterals; (2) As shown in the figure, triangle ABC is rotated 60° clockwise around vertex B to obtain triangle DBE. Connect AD and BC, with ∠DCB=30°. Prove that DC^{2}+ BC^{2}=AC^{2}, indicating that the quadrilateral is a Pythagorean quadrilateral.

Pythagorean Quadrilateral

<image>Point $A,B,C,D,$ and $E$ are equally spaced on a minor arc of a circle. Points $E,F,G,H,I$ and $A$ are equally spaced on a minor arc of a second circle with center $C$ as shown in the figure below. The angle $\angle ABD$ exceeds $\angle AHG$ by $12^\circ$. Find the degree measure of $\angle BAG$.

58

<image>The square $A B C D$ has area 80. The points $E, F, G$ and $H$ are on the sides of the square and $\mathrm{AE}=\mathrm{BF}=\mathrm{CG}=\mathrm{DH}$. How big is the area of the grey part, if $\mathrm{AE}=3 \times \mathrm{EB}$?

25

<image>ABCD conforms a parallelogram. FEGH forms a square. FHI is an isosceles triangle, with FH equalling HI. Can you specify the length of the base FI in the isosceles triangle FHI?

8*√(3)

<image>EFG results in an equilateral triangle. Square GEHI has a circle inscribed. What is the total area of the circle?

π/4

<image>In the figure shown, all quadrilaterals are squares, and all triangles are right-angled triangles. The hypotenuse of the largest right-angled triangle is 9 cm. The sum of the areas of the four shaded squares is ____ cm². ______

81

<image>As shown in the figure, if $ l_1 \parallel l_2 $ and $ \angle 1 = 50^\circ $, then $ \angle 2 = \_\_\_^\circ $.

130

<image>CDE describes an equilateral triangle. Side GH shapes into an equilateral triangle. IGJ is an isosceles triangle, with IG equalling GJ. Can you tell me the length of the base IJ in the isosceles triangle IGJ?

80*√(6 - 3*√(2))/3

<image>Side EF takes the shape of an equilateral triangle. Square FGHI has an inscribed circle. How do you find the perimeter of inscribed circle?

65*π

<image>Side DE extending into an equilateral triangle shape beyond the rectangle. Side GH developing into an equilateral triangle outside. What is the area of the entire shape HIJK?

540*√(2)

<image>As shown in the figure, in rectangle $ABCD$, point $E$ is on side $AB$. △$EBC$ is folded along the line $CE$, so that point $B$ falls on point $B′$ on side $AD$. When the folded figure is unfolded, if the perimeter of △$AB′E$ is 4 cm and the perimeter of △$B′DC$ is 11 cm, then the length of $B′D$ is _________ cm.

3.5

<image> As shown in the figure, the side length of square ABCD is 3, and the side length of square AEFG is 4. Given that $ S_1 = S_2 $, $ S_3 = S_4 $, and $ S_5 = S_6 $, what is the area of square DEHK?

25

<image>Side EF is shaped into an equilateral triangle. In shape FGHI, what is the length of IH?

7

<image> As shown in the figure, AB ∥ CD, CD ∥ EF, ∠BCE = 90°. Find the degree measure of ∠E - ∠B.

90°

<image>In triangle $ABC$, $AB = AC = 5$ and $BC = 6$. Let $O$ be the circumcenter of triangle $ABC$. Find the area of triangle $OBC$.

$\frac{21}{8}$

<image> As shown in the figure, points A, O, and B are on the same straight line, and OC bisects ∠AOD. ∠AOC + ∠EOB = 90°. (1) Find the measure of ∠COE. (2) Determine the relationship between ∠DOE and ∠EOB, and provide reasoning.

90°

<image>Side EF forming an outward equilateral triangle beyond the rectangle. Square GEHI contains an inscribed circle. Please provide the length of IH in shape GEHI.

21

<image> On a lake surface as calm as a mirror, there stands a beautiful red lotus, which is $$1$$ meter above the water. A strong gust of wind blows, causing the red lotus to be swept to the side until the flower is level with the water surface. Given that the horizontal distance moved by the red lotus is $$2$$ meters, what is the depth of the water here?

1.5

<image>The two triangles in the figure are both isosceles right triangles. Find the area of the shaded region in the figure.

24.5

<image> Cut the following image into 4 small squares. Find the perimeter and side length of each small square.

2

<image>ABC is an isosceles triangle with AB having the same length as BC. CBD is an isosceles triangle in which CB equals BD in length. DBEF is identified as a parallelogram. What is the overall area measurement of the shape DBEF?

1470

<image>Side CD converting into an equilateral triangle outside. ECF is an isosceles triangle, with EC equalling CF. What is the full perimeter of FCGH?

8*√(2) + 14

<image>ACDE forms a square. DCF is an isosceles triangle, with DC equalling CF. How much is the area of the entire shape FCG?

5400*√(3)

<image>ACD is an isosceles triangle with AC and CD being equal in length. DCE is defined as a sector. Side FG continues into a semi-circle within the rectangle. Please provide the length of FG in shape ECFG.

√(3)*π/3

<image>ABCD is geometrically a square. EBFG takes the form of a parallelogram. GFHI is geometrically a square. In shape GFHI, how long is IH?

3*√(3)

<image>ABCD is a parallelogram. DCE is an isosceles triangle in which DC and CE are equal. How would you determine the perimeter of ECF?

22*√(3) + 66

<image>DCEF is geometrically a square. DFGH is geometrically a square. How do you calculate the area of the shape DFGH?

100

<image>DCEF is identified as a parallelogram. Side GH takes the shape of an equilateral triangle. What is the entire perimeter length of FEGIH?

440

<image>As shown in the figure , the perimeter of this rectangle is ____ decimeters.

10

<image>ABC is an isosceles triangle, with AB equalling BC. CBDE is a square. EDF is an isosceles triangle with ED and DF being equal in length. How do you find the area of the entire shape EDF?

18

<image>ABCD is geometrically a parallelogram. DCEF is geometrically a square. DFGH is geometrically a square. What is the overall perimeter of DFGH?

52

<image> As shown in the figure, points A, B, and C are on circle O, AC ∥ OB, and ∠BOC = 40°. Then ∠ABO = ____ degrees.

20

<image>As shown in the figure, in the circle ⊙O, OA ∥ BC, and ∠ACB = 20°. Find ∠1 = ____.

60

<image>ACD is an isosceles triangle where the lengths of AC and CD are equal. What is the full area of the shape DCE?

4704*√(3)

<image>ABCD is geometrically a square. EFG is geometrically an equilateral triangle. HEIJ is a parallelogram. How would you determine the perimeter of HEIJ?

5

<image>According to the diagram, a large rectangle is formed by 6 squares. Given that the smallest square has an area of 1 cm², what is the area of the large rectangle in cm²?

143

<image>As shown in the figure, in trapezoid ABCD, AD∥BC, AB=a, DC=b. The perpendicular bisector of side DC, EF, intersects side BC at E, and E is the midpoint of BC. Additionally, DE∥AB. Find the perimeter of trapezoid ABCD.

4a+b

<image>As shown in the image, a rectangular piece of paper has a width AB = 8 cm and a length BC = 10 cm. The paper is folded so that vertex D falls on point F on edge BC (the fold line is AE). Find the length of EC.

3

<image>ABCD is geometrically a square. ADE is an isosceles triangle where AD = DE. EDF is an isosceles triangle with ED and DF being equal in length. I need to know the length of the base EF in the isosceles triangle EDF.

56

<image>As shown in the , an isosceles right triangle $$ABC$$ is rotated clockwise around point $$C$$ on a horizontal table surface to position $$A'B'C$$, such that points $$A$$, $$C$$, and $$B'$$ are collinear. The measure of the angle of rotation is ______ .

135

<image>EFG constitutes an equilateral triangle. How do you find the area of the entire shape ADEGF?

4704*√(3) + 4592*√(6)

<image>As shown in the figure, fold one side of the rectangle $$ABCD$$ along $$AD$$ such that point $$D$$ falls on point $$F$$ on side $$BC$$. Given that the crease $$AE=5 \sqrt {5}cm$$ and $$\tan ∠EFC= \dfrac {3}{4}$$, what is the perimeter of the rectangle $$ABCD$$?

36

<image>As shown in the image, it is known that AB∥CD, BF bisects ∠ABE, DF bisects ∠CDE, and ∠BFD = 140°. The measure of ∠BED is ______.

80

<image>As shown in the figure, AB is the diameter of circle O, and points C and D are on circle O. If ∠C = 20°, then the measure of ∠ABD is (__)

70

<image> When hosting a TV show, the host stands at the golden section point of the stage to appear the most natural and graceful. As shown in the figure, if the stage AB is 20 meters long and the host is currently standing at point A, how many meters should the host walk from point A to appear the most natural and graceful? (Result accurate to 0.1 meters)______.

7.6

<image>Known from the : As shown in the figure, ∠BOC = 2∠AOB, OD bisects ∠AOC, and ∠BOD = 20°, find the degree of ∠AOB.

40

<image>ABC is an isosceles triangle, having AB and BC of the same length. CBD is in the form of a sector. DBE is an isosceles triangle, with DB equalling BE. How long is the perimeter of EBFG?

38

<image>Carefully observe the angles in the image.

1