Sicong/RL_toy_math_3 · Datasets at Hugging Face

images images listlengths 1 1	problem stringlengths 7 3.42k	answer stringlengths 1 40
	<image>If the arc length of the EAB sector is 12.85, the arc length of the FAG sector is 15.42 and the angle FAG is vertical to BAE, compute the diagonal of the ABCD rectangle. Assume $\pi=3.14$. Round computations to 2 decimal places.	18.48
	<image>ABC conforms a sector. DEF results in an equilateral triangle. Side GH develops into an equilateral triangle. HIJK is a square. Please provide the length of JI in shape HIJK.	98
	<image>DEF is an isosceles triangle where the length of DE equals the length of EF. FEGH is a parallelogram. What is the side length HG in the shape FEGH?	22*√(3)/3
	<image>ABC is identified as a sector. CBDE takes the form of a parallelogram. EDF is defined as a sector. How do you find the perimeter of FDGH?	46
	<image> The technology team is making an airplane model. The plane view of the wings is made up of two identical trapezoids. With one side of the wing (as shown in the figure), what is its area? (Unit: cm)	360
	<image>Find the area of the shaded region in the figure.	2\pi + 12
	<image> As shown in the figure, there is a rectangular grassland with a length of 16m and a width of 10m. Two paths have been built in the middle, one is parallelogram-shaped and the other is rectangular. What is the area of the grassland now in square meters?	112
	<image>As shown in the figure, there is an iron tower AB at the top of a slope, and under sunlight, the shadow of the tower DE is cast on the slope. Given that the base width of the tower CD is 14m, the length of the tower's shadow DE is 36m. Xiaoming and Xiaohua both have a height of 1.6m. Xiaoming stands at point E, and his shadow is also on the slope. Xiaohua stands at the foot of the slope along the direction of DE, and his shadow is on the flat ground. The lengths of their shadows are 4m and 2m respectively. Find the height AB of the tower.	20
	<image>As shown in the figure, AB is the diameter of circle O. With B as the center and BO as the radius, an arc is drawn intersecting circle O at points C and D. Then the measure of ∠BCD is (___ ).	30
	<image>ABC is identified as a sector. CBD is geometrically a sector. EFG exemplifies an equilateral triangle. Please provide the length of GE in shape DBEGF.	65
	<image>Calculate the perimeter and area of each of the following shapes.	198
	<image>As shown in the figure, OD bisects ∠AOC, ∠AOB = 3∠COD, ∠BOC = 4∠AOD. Find the degree measure of ∠AOB.	120
	<image>FEGH is a square. What is the area of the entire shape FEGH?	57624
	<image>ABC takes the form of a sector. CDEF is identified as a parallelogram. What is the length of FG in shape FEG?	4/3
	<image>I surrounded the wooden circle (see picture) using $a \mathrm{~cm}$ of thread. After that I surrounded by thread the wooden square –– $b \mathrm{~cm}$ of thread was enough for that. How much thread (in $\mathrm{cm}$ ) would be enough to surround the three wooden circles without moving them? A. 3a B. 2a+b C. a+2b D. 3b E. a+b	E. a+b
	<image>As shown in the figure, in △ABC, AD is the altitude, AE and BF are angle bisectors that intersect at point G. AD and BF intersect at point H. ∠BAC=50°, ∠C=70°. Then ∠AHB= ______ .	120
	<image>CDE describes an equilateral triangle. DEFG is geometrically a parallelogram. GFHI is a square. In shape IHJK, how long is KJ?	13
	<image>ABCD is geometrically a parallelogram. FEGH is a square. Arc IJ is shaped as a semi-circle. How long is IJ in shape FHIJ?	π/2
	<image>Two congruent squares, $ABCD$ and $PQRS$, have side length $15$. They overlap to form the $15$ by $25$ rectangle $AQRD$ shown. What percent of the area of rectangle $AQRD$ is shaded?	20
	<image>DCE is an isosceles triangle where the sides DC and CE are identical in length. Side FG morphs into an equilateral triangle. Side IJ developing into an equilateral triangle outside. How can we determine the area of the entire shape GHIKJ?	845*√(3)/4
	<image> Calculate the shaded area. R = 3 dm; r = 1 dm.	25.12
	<image>ABC is an isosceles triangle where AB = BC. CDE is an isosceles triangle where CD and DE are of equal length. What is the length of ED in shape CDE?	7*√(2)
	<image>As shown in the figure, in the parallelogram \(ABCD\), an arc is drawn with center \(A\) and radius equal to the length of \(AB\), intersecting \(AD\) at point \(F\). Then, arcs are drawn with centers at points \(B\) and \(F\) with radius greater than \(\frac{1}{2} BF\), intersecting at point \(P\). Segment \(AP\) is extended to intersect \(BC\) at point \(E\). Connect \(EF\) and \(AE\), and \(BF\) intersects these segments at point \(O\). If the perimeter of quadrilateral \(ABEF\) is 40 and \(BF = 10\), find \(\angle ABC =\) _____. _	120^\circ
	<image>Given: As shown in the diagram, AB ∥ CD. The line EF intersects AB and CD at points E and F, respectively. EG bisects ∠BEF. If ∠2 = 65°, then ∠1 = ____°.	50
	<image>ABCD is geometrically a parallelogram. DCE is an isosceles triangle where the length of DC equals the length of CE. Side FG extends into an equilateral triangle outside the rectangle. Side IJ develops into an equilateral triangle. Could you specify the length of JK in shape GHIKJ?	14
	<image>ABCD is a parallelogram. ECF is an isosceles triangle where EC is the same length as CF. Side GH extends inward as an equilateral triangle within the rectangle. What is the total length of the perimeter of FCGIH?	4 + 3*√(3)
	<image>Side CD forming an equilateral triangle. What is the measurement of the total area of GFH?	1225*√(3)/2
	<image> (2014 Spring•End of term in Daiyue District) As shown in the figure, BA = BC, EB ∥ AC, and ∠A = 55°, then the measure of ∠DBE is ____.	55
	<image>ABCD is geometrically a square. IJK is in the shape of an equilateral triangle. Can you tell me the length of JK in shape HGIKJ?	77
	<image>Side CD being an equilateral triangle. ECF conforms a sector. GHI is geometrically an equilateral triangle. What is the area size of the entire shape FCGIH?	3*√(3)/4
	<image>ABC is in the form of a sector. CBD is an isosceles triangle with CB and BD being equal in length. DBE takes the form of a sector. EBFG is a square. What's the area of the entire shape EBFG?	8100
	<image>DEF is classified as an equilateral triangle. EFGH forms a square. Side IJ extends to create a semi-circle inside the rectangle. How much is the area of the entire shape GFIJ?	-147π/8 + 189√(2)
	<image>DCE is an isosceles triangle where DC and CE are of equal length. ECF is geometrically a sector. Can you calculate the perimeter of ECF?	π/3 + 4
	<image>Guangming Elementary School has renovated the playground and built a circular track. The shaded part in the image is the activity area. ① Please calculate the area of the shaded activity area. ② If you run one lap around the track, how many meters will you need to run?	4856
	<image> In the triangle △ABC shown in the figure, AD is the altitude on side BC, AE is the angle bisector of ∠BAC, ∠B = 40°, ∠C = 70°, ∠DAE = ____________.	15°
	<image>ABC conforms a sector. CDE is geometrically a sector. What is the measurement of the perimeter of EDFG?	16*√(3) + 36
	<image>【Problem Statement】As shown in the figure, the side length of each small square in the square grid is equal. The three vertices A, B, and C of △ABC are all on the grid points. If the grid point D is on the circumcircle of △ABC, then the number of points D in the figure that meet the conditions is _ ▲ _ (point D does not coincide with points A, B, or C).	5
	<image>Side DE morphs into an equilateral triangle. FDGH conforms a square. In the shape FDGH, what is the measurement of GD?	20*√(2)
	<image> The diagram below is a triangle. Arcs are drawn with a radius of 2 cm centered at each of its vertices. Find the area of the shaded region.	6.28
	<image>ACDE is geometrically a square. Can you tell me the perimeter of EDFG?	2√(6) + 4√(3)
	<image>CDE describes an equilateral triangle. Arc GH constitutes a semi-circle. Could you specify the length of GH in shape FCGH?	91*π/2
	<image> As shown in the figure, △ABC is folded along EF, and the folded figure is as shown in the picture. If ∠A = 60° and ∠1 = 96°, then the measure of ∠2 is ______.	24
	<image>ABC is an isosceles triangle where the length of AB equals the length of BC. Side DE extends into an equilateral triangle outside the rectangle. Side GH converts into a semi-circle. What is the arc length GH in the shape EFGH?	π
	<image>ABC is an isosceles triangle, having AB and BC of the same length. Side DE stretching into an equilateral triangle outside the rectangle. HGIJ forms a square. Can you calculate the perimeter of HGIJ?	4
	<image>ABCD is identified as a parallelogram. DCEF is a square. Side GH growing into a semi-circle outside the rectangle. What is the area size of the entire shape ECGH?	648π + 2592√(3)
	<image>Side CD builds into an equilateral triangle. ECFG is a square. FCH takes the form of a sector. Side IJ is extended into a semi-circular shape inside the rectangle. What is the length of IJ in shape HCIJ?	4*π
	<image>As shown in the diagram, points A, O, and E are collinear, and ∠AOB=40°, ∠EOD=30°, with OD bisecting ∠COE. Find the measure of ∠COB. (10 points) __	80
	<image>Can you calculate the area of the billboard in square centimeters (cm²) based on its length and width as shown in the image? How many square decimeters (dm²) is it?	65
	<image>ABC is in the form of a sector. CBDE is a parallelogram. EDF is an isosceles triangle, having ED and DF of the same length. What would the perimeter of FDGH be?	4
	<image> As shown in the figure, in △ABC, ∠ACB = 90°, D is a point on the extension line of BC, DF is perpendicular to AB at F, and DF intersects AC at E. How many right-angled triangles are there in the figure? Which angles are equal to ∠A? Which angles are complementary to ∠A? Please list them separately.	4
	<image>As shown in the figure, add a condition: _____________ , to make △ADE ~ △ACB (write one condition only) __	∠ADE = ∠ACB
	<image>FGH is geometrically an equilateral triangle. How would you calculate the perimeter of ECFHG?	17
	<image>ACDE is a parallelogram. EDF is an isosceles triangle where the sides ED and DF are identical in length. Square FDGH includes a circle inscribed within. How long is FH in shape FDGH?	10
	<image>ABC forms a sector. CBD takes on a sector shape. DBEF takes the form of a parallelogram. How long is HG in shape FEGH?	1
	<image>Side CD shaping into an equilateral triangle beyond the rectangle. DEFG is a parallelogram. IJK is shaped as an equilateral triangle. What is the total area of the shape HFIKJ?	9*√(3)/4
	<image>In the following groups, ______ are the shapes that are the same, and ______ are the shapes that are different.	(3)(5) (1)(2)(4)(6)
	<image>As shown in the figure, the lower right corner of the quadrilateral paper $$ABCD$$ is folded inward to form $$\triangle PC′R$$, where $$∠B=120^{\circ}$$ and $$∠D=40^{\circ}$$, such that $$C′P \parallel AB$$ and $$RC′ \parallel AD$$. Then, $$∠C =$$ ______.	100°
	<image> As shown in the figure, the diagonals AC and BD of trapezoid ABCD intersect at point O. If \( S_{△AOD} : S_{△ACD} = 1 : 4 \), then \( S_{△AOD} : S_{△BOC} = \_\_\_\_ \).	1:9
	<image>(2014 Spring, Wuxi Final Exam) As shown in the figure, square ABCD with a side length of 3 cm is first translated 1 cm to the right and then 1 cm upward, resulting in square EFGH. The area of the shaded region is ____.	4
	<image>Side DE converts into an equilateral triangle. Square EFGH encompasses an inscribed circle. What would the perimeter of inscribed circle be?	72√(3)π
	<image>(2012 Autumn, Jiaxing School-level Midterm) As shown in the figure, CD is the diameter of circle O, and point C is on the minor arc subtended by chord AB. If point C is the midpoint of arc AB, then the measure of ∠AMD is ____.	90
	<image>(2014 Autumn • Final Exam at Funan County) Xiaoming accidentally broke a triangular piece of glass into four pieces as shown in the diagram (labeled 1, 2, 3, and 4). In your opinion, which one of these pieces should be taken along to be able to match a piece of triangular glass of the original size? You should take piece ____, explain the reason.	2
	<image>DCEF is geometrically a square. What is the measurement of DG in the shape DFG?	48*√(3)
	<image>As shown in the figure, in the rectangular coordinate system, points \( A \) and \( C \) are on the \( x \)-axis and \( y \)-axis respectively. \( CB \parallel OA \), \( CB=8 \), \( OC=8 \), \( OA=16 \). (1) Directly write down the coordinates of points \( A \), \( B \), and \( C \) and the area of quadrilateral \( OABC \). (2) The moving point \( P \) starts from the origin \( O \) and moves towards \( A \). It stops when the line \( PC \) divides the quadrilateral \( OABC \) into two parts of equal area. Find the coordinates of point \( P \). (3) Under the condition of (2), is there a point \( Q \) on the \( y \)-axis such that the area of triangle \( CPQ \) is equal to the area of quadrilateral \( OABC \)? If such a point exists, find the coordinates of point \( Q \); if not, please explain why.	96
	<image>As shown in the figure, in the rectangle $$ABCD$$, the diagonals $$AC$$ and $$BD$$ intersect at point $$O$$, $$AE \perp BD$$ at point $$E$$, and $$∠AOB = 50^{\circ}$$. What is the degree measure of $$∠BAE$$? ______ .	25
	<image>(2014 Autumn • Chiping Middle School Midterm) As shown in the figure, point P is inside ∠AOB. Points M and N are the symmetric points of point P with respect to AO and BO, respectively. Points E and F are on the sides OA and OB, respectively. If the perimeter of triangle △PEF is 15, then the length of MN is ____.	15
	<image>As shown in the figure: AB=AC, ∠A=50°, point O is a point inside △ABC, and ∠OBC=∠ACO, then ∠BOC=__ .__	115°
	<image>ABCD conforms a square. CBE forms a sector. GFHI is geometrically a parallelogram. What would the perimeter of GFHI be?	6*√(3) + 14
	<image>As shown in the picture, a street lamp is 8 meters above the ground, and Xiao Ming, who is 1.6 meters tall, is standing at point A, which is 20 meters from the base of the lamp (point O). The length of Xiao Ming's shadow is (___) meters.	5
	<image> As shown in the figure, given that $\triangle ABC \cong \triangle BAE$, $\angle ABE = 60^{\circ}$, and $\angle E = 92^{\circ}$, then the measure of $\angle ABC$ is ______ degrees.	28
	<image>As shown in the figure, given that ∠1 = 30°, then ∠2 = ____, ∠3 = ____.	150
	<image>Side DE stretching into an equilateral triangle outside the rectangle. FDGH conforms a square. How much is the perimeter of FDGH?	208*√(3)/3
	<image>FEG is a sector. What is the measurement of the perimeter of FEG?	45*π/2 + 90
	<image>As shown in the figure, sunlight enters the classroom through the window, with the shadow of the window frame AB on the ground being DE=1.8m. The distance from the bottom edge of the window to the ground BC is 1m, and EC is 1.2m. What is the height of the window AB in meters?	1.5
	<image> As shown in the figure, $$∠B=30^{\circ}$$. If $$AB \parallel CD$$ and $$CB$$ bisects $$∠ACD$$, then $$∠ACD =$$ ______ degrees.	60
	<image>(2012 Spring • Hejiang County School Midterm) As shown in the , in △ABC, AE is the median, AD is the angle bisector, and AF is the altitude, then: (1) ∠BAC = 2____; (2) BC = 2____; (3) ____ = 90°.	\angle BAD
	<image>As shown in the figure, in triangle ABC, ∠A=30°, ∠C=60°, DC=1cm, DE ⊥ bisects AB. Then AD=______cm.	1
	<image> (2015 Autumn • Cangnan County Final) As shown in the figure, lines AB and CD intersect at point O, and OE bisects ∠BOD. If ∠AOE = 144°, then the degree measure of ∠AOC is ____.	72
	<image> As shown in the figure, D is a point on side AB of △ABC. Through point D, draw DE ∥ BC, intersecting AC at point E. Through point E, draw EF ∥ AB, intersecting BC at point F. Then, the number of triangles similar to △ABC in the figure is ____. (excluding △ABC)	2
	<image>As shown in the figure, the side length of square $$ABCD$$ is $$1$$. Points $$P$$ and $$Q$$ are on sides $$AB$$ and $$DA$$ respectively. When the perimeter of $$\triangle APQ$$ is $$2$$, find the measure of $$∠PCQ$$.	45^\circ
	<image>ABC is defined as a sector. CBD is defined as a sector. What is the total length of the perimeter of DBE?	9*√(2) + 18
	<image>As shown in the , line AB and CD intersect at point O. If ∠1 + ∠2 = 100°, then ∠BOC is equal to ______.	130
	<image>If the length of the AB side is 20, the length of the AD side is $2x + 5$, the length of the AC side is $x + 3.92$, the degree of the CAD angle is 25 and the degree of the CDA angle is 30, compute the degree of the CBA angle. Round computations to 2 decimal places and round the value of the variable "x" to the nearest natural number.	23.58
	<image>ABCD takes the form of a parallelogram. DCE takes the form of a sector. What is the measurement of EF in the shape ECF?	21*√(2)
	<image>(Autumn 2015, Midterm Examination in Binhu District) As shown in the figure, in quadrilateral ABCD, AD ∥ BC, ∠B = 90°, AB = AD = 4 cm, and BC = 7 cm. Now, a paper in the shape of quadrilateral ABCD is to be cut to obtain an isosceles triangle with leg length of 3 cm (Requirements: One vertex of the isosceles triangle coincides with one vertex of quadrilateral ABCD, and the other two vertices lie on the sides of quadrilateral ABCD). Then, there are ____ possible values for the length of the base of the cut isosceles triangle.	4
	<image>ABC is an isosceles triangle with AB and BC being equal in length. CBDE conforms a square. EDFG is a parallelogram. Please provide the length of GF in shape EDFG.	7
	<image>Point P is a moving point on the ellipse =1 with its major axis on the x-axis. F_{1} and F_{2} are the two foci of the ellipse, and the semi-focal distance is c. Then, the maximum value of \|PF_{1}\|•\|PF_{2}\| is (__)	a^2
	<image>Side CD is designed as an equilateral triangle. ECF takes on a sector shape. GHI defines an equilateral triangle. Square HIJK has an inner inscribed circle. What is the overall perimeter of circle?	3*π
	<image>Side CD expanding into an equilateral triangle beyond the rectangle. DEFG conforms a square. GFHI is geometrically a square. Could you specify the length of HF in shape GFHI?	18
	<image>As shown in the figure, in right triangle ABC, ∠C=90°, AD bisects ∠BAC and intersects BC at D, with BD:DC=5:4 and BC=18 cm. Then the distance from D to side AB is ___ cm.	8
	<image>As shown in the figure, a right-angled triangular piece of paper is folded so that points B and C overlap, with the crease being DE. Connect DC. If AC = 6 cm, ∠ACB = 90°, ∠B = 30°, then the perimeter of △ADC is ______cm.	18
	<image>As shown in the figure, points C and F are on BE, ∠1 = ∠2, and BC = EF. Please add one more condition: ___ (just one), to make △ABC ≌ △DEF.	∠B = ∠E
	<image>ABCD takes the form of a parallelogram. How much is the perimeter of ECFG?	72 + 54*√(3)
	<image>As shown in the figure, a rectangular piece of paper is folded along EF, causing points D and C to fall on positions \( D_{1} \) and \( C_{1} \) respectively. If ∠EFB=65°, then ∠BFC_{1}=__▲__°.	50
	<image>Explore: (1) As shown in Figure ①, what is the relationship between ∠1 + ∠2 and ∠B + ∠C? Why? (2) Folding triangle △ABC along DE in Figure ①, we get Figure ②. Fill in the blank: ∠1 + ∠2 ______ ∠B + ∠C (fill in with “>”, “<”, or “=”), and when ∠A = 40°, ∠B + ∠C + ∠1 + ∠2 = ______; (3) As shown in Figure ③, which is obtained by folding △ABC along DE in Figure ①, if ∠A = 30°, then x + y = 360° - (∠B + ∠C + ∠1 + ∠2) = 360° - ______ = ______. Guess the relationship between ∠BDA + ∠CEA and ∠A: ______.	=
	<image> As shown in the figure, $$\triangle ABC$$ ≌ $$\triangle DEF$$, $$∠A = 25^{\circ}$$, $$∠B = 65^{\circ}$$, $$BF = 3cm$$. Find the measure of $$∠DFE$$ and the length of $$EC$$.	90°
	<image>In the figure, in $$\triangle ABC$$, $$∠ACB=90^{\circ}$$, $$D$$, $$E$$, and $$F$$ are the midpoints of sides $$AB$$, $$AC$$, and $$BC$$ respectively. Given that $$CD=3$$, find the length of $$EF$$.	3
	<image>According to the figure, square ABCD with a side length of 3 cm is first translated 1 cm to the right and then 1 cm upwards to obtain square EFGH. The area of the shaded region is _________ square centimeters.	4
	<image>ABCD is a square. ECF is an isosceles triangle in which EC equals CF in length. FCG takes the form of a sector. In shape FCG, what is the length of GC?	9
	<image>DCEF is geometrically a square. What is the overall area measurement of the shape DFGH?	4*√(2)

<image>If the arc length of the EAB sector is 12.85, the arc length of the FAG sector is 15.42 and the angle FAG is vertical to BAE, compute the diagonal of the ABCD rectangle. Assume $\pi=3.14$. Round computations to 2 decimal places.

18.48

<image>ABC conforms a sector. DEF results in an equilateral triangle. Side GH develops into an equilateral triangle. HIJK is a square. Please provide the length of JI in shape HIJK.

98

<image>DEF is an isosceles triangle where the length of DE equals the length of EF. FEGH is a parallelogram. What is the side length HG in the shape FEGH?

22*√(3)/3

<image>ABC is identified as a sector. CBDE takes the form of a parallelogram. EDF is defined as a sector. How do you find the perimeter of FDGH?

46

<image> The technology team is making an airplane model. The plane view of the wings is made up of two identical trapezoids. With one side of the wing (as shown in the figure), what is its area? (Unit: cm)

360

<image>Find the area of the shaded region in the figure.

2\pi + 12

<image> As shown in the figure, there is a rectangular grassland with a length of 16m and a width of 10m. Two paths have been built in the middle, one is parallelogram-shaped and the other is rectangular. What is the area of the grassland now in square meters?

112

<image>As shown in the figure, there is an iron tower AB at the top of a slope, and under sunlight, the shadow of the tower DE is cast on the slope. Given that the base width of the tower CD is 14m, the length of the tower's shadow DE is 36m. Xiaoming and Xiaohua both have a height of 1.6m. Xiaoming stands at point E, and his shadow is also on the slope. Xiaohua stands at the foot of the slope along the direction of DE, and his shadow is on the flat ground. The lengths of their shadows are 4m and 2m respectively. Find the height AB of the tower.

20

<image>As shown in the figure, AB is the diameter of circle O. With B as the center and BO as the radius, an arc is drawn intersecting circle O at points C and D. Then the measure of ∠BCD is (___ ).

30

<image>ABC is identified as a sector. CBD is geometrically a sector. EFG exemplifies an equilateral triangle. Please provide the length of GE in shape DBEGF.

65

<image>Calculate the perimeter and area of each of the following shapes.

198

<image>As shown in the figure, OD bisects ∠AOC, ∠AOB = 3∠COD, ∠BOC = 4∠AOD. Find the degree measure of ∠AOB.

120

<image>FEGH is a square. What is the area of the entire shape FEGH?

57624

<image>ABC takes the form of a sector. CDEF is identified as a parallelogram. What is the length of FG in shape FEG?

4/3

<image>I surrounded the wooden circle (see picture) using $a \mathrm{~cm}$ of thread. After that I surrounded by thread the wooden square –– $b \mathrm{~cm}$ of thread was enough for that. How much thread (in $\mathrm{cm}$ ) would be enough to surround the three wooden circles without moving them? A. 3a B. 2a+b C. a+2b D. 3b E. a+b

E. a+b

<image>As shown in the figure, in △ABC, AD is the altitude, AE and BF are angle bisectors that intersect at point G. AD and BF intersect at point H. ∠BAC=50°, ∠C=70°. Then ∠AHB= ______ .

120

<image>CDE describes an equilateral triangle. DEFG is geometrically a parallelogram. GFHI is a square. In shape IHJK, how long is KJ?

13

<image>ABCD is geometrically a parallelogram. FEGH is a square. Arc IJ is shaped as a semi-circle. How long is IJ in shape FHIJ?

π/2

<image>Two congruent squares, $ABCD$ and $PQRS$, have side length $15$. They overlap to form the $15$ by $25$ rectangle $AQRD$ shown. What percent of the area of rectangle $AQRD$ is shaded?

20

<image>DCE is an isosceles triangle where the sides DC and CE are identical in length. Side FG morphs into an equilateral triangle. Side IJ developing into an equilateral triangle outside. How can we determine the area of the entire shape GHIKJ?

845*√(3)/4

<image> Calculate the shaded area. R = 3 dm; r = 1 dm.

25.12

<image>ABC is an isosceles triangle where AB = BC. CDE is an isosceles triangle where CD and DE are of equal length. What is the length of ED in shape CDE?

7*√(2)

<image>As shown in the figure, in the parallelogram $ABCD$, an arc is drawn with center $A$ and radius equal to the length of $AB$, intersecting $AD$ at point $F$. Then, arcs are drawn with centers at points $B$ and $F$ with radius greater than $\frac{1}{2} BF$, intersecting at point $P$. Segment $AP$ is extended to intersect $BC$ at point $E$. Connect $EF$ and $AE$, and $BF$ intersects these segments at point $O$. If the perimeter of quadrilateral $ABEF$ is 40 and $BF = 10$, find $\angle ABC =$ _____. _

120^\circ

<image>Given: As shown in the diagram, AB ∥ CD. The line EF intersects AB and CD at points E and F, respectively. EG bisects ∠BEF. If ∠2 = 65°, then ∠1 = ____°.

50

<image>ABCD is geometrically a parallelogram. DCE is an isosceles triangle where the length of DC equals the length of CE. Side FG extends into an equilateral triangle outside the rectangle. Side IJ develops into an equilateral triangle. Could you specify the length of JK in shape GHIKJ?

14

<image>ABCD is a parallelogram. ECF is an isosceles triangle where EC is the same length as CF. Side GH extends inward as an equilateral triangle within the rectangle. What is the total length of the perimeter of FCGIH?

4 + 3*√(3)

<image>Side CD forming an equilateral triangle. What is the measurement of the total area of GFH?

1225*√(3)/2

<image> (2014 Spring•End of term in Daiyue District) As shown in the figure, BA = BC, EB ∥ AC, and ∠A = 55°, then the measure of ∠DBE is ____.

55

<image>ABCD is geometrically a square. IJK is in the shape of an equilateral triangle. Can you tell me the length of JK in shape HGIKJ?

77

<image>Side CD being an equilateral triangle. ECF conforms a sector. GHI is geometrically an equilateral triangle. What is the area size of the entire shape FCGIH?

3*√(3)/4

<image>ABC is in the form of a sector. CBD is an isosceles triangle with CB and BD being equal in length. DBE takes the form of a sector. EBFG is a square. What's the area of the entire shape EBFG?

8100

<image>DEF is classified as an equilateral triangle. EFGH forms a square. Side IJ extends to create a semi-circle inside the rectangle. How much is the area of the entire shape GFIJ?

-147*π/8 + 189*√(2)

<image>DCE is an isosceles triangle where DC and CE are of equal length. ECF is geometrically a sector. Can you calculate the perimeter of ECF?

π/3 + 4

<image>Guangming Elementary School has renovated the playground and built a circular track. The shaded part in the image is the activity area. ① Please calculate the area of the shaded activity area. ② If you run one lap around the track, how many meters will you need to run?

4856

<image> In the triangle △ABC shown in the figure, AD is the altitude on side BC, AE is the angle bisector of ∠BAC, ∠B = 40°, ∠C = 70°, ∠DAE = ____________.

15°

<image>ABC conforms a sector. CDE is geometrically a sector. What is the measurement of the perimeter of EDFG?

16*√(3) + 36

<image>【Problem Statement】As shown in the figure, the side length of each small square in the square grid is equal. The three vertices A, B, and C of △ABC are all on the grid points. If the grid point D is on the circumcircle of △ABC, then the number of points D in the figure that meet the conditions is _ ▲ _ (point D does not coincide with points A, B, or C).

5

<image>Side DE morphs into an equilateral triangle. FDGH conforms a square. In the shape FDGH, what is the measurement of GD?

20*√(2)

<image> The diagram below is a triangle. Arcs are drawn with a radius of 2 cm centered at each of its vertices. Find the area of the shaded region.

6.28

<image>ACDE is geometrically a square. Can you tell me the perimeter of EDFG?

2*√(6) + 4*√(3)

<image>CDE describes an equilateral triangle. Arc GH constitutes a semi-circle. Could you specify the length of GH in shape FCGH?

91*π/2

<image> As shown in the figure, △ABC is folded along EF, and the folded figure is as shown in the picture. If ∠A = 60° and ∠1 = 96°, then the measure of ∠2 is ______.

24

<image>ABC is an isosceles triangle where the length of AB equals the length of BC. Side DE extends into an equilateral triangle outside the rectangle. Side GH converts into a semi-circle. What is the arc length GH in the shape EFGH?

π

<image>ABC is an isosceles triangle, having AB and BC of the same length. Side DE stretching into an equilateral triangle outside the rectangle. HGIJ forms a square. Can you calculate the perimeter of HGIJ?

4

<image>ABCD is identified as a parallelogram. DCEF is a square. Side GH growing into a semi-circle outside the rectangle. What is the area size of the entire shape ECGH?

648*π + 2592*√(3)

<image>Side CD builds into an equilateral triangle. ECFG is a square. FCH takes the form of a sector. Side IJ is extended into a semi-circular shape inside the rectangle. What is the length of IJ in shape HCIJ?

4*π

<image>As shown in the diagram, points A, O, and E are collinear, and ∠AOB=40°, ∠EOD=30°, with OD bisecting ∠COE. Find the measure of ∠COB. (10 points) __

80

<image>Can you calculate the area of the billboard in square centimeters (cm²) based on its length and width as shown in the image? How many square decimeters (dm²) is it?

65

<image>ABC is in the form of a sector. CBDE is a parallelogram. EDF is an isosceles triangle, having ED and DF of the same length. What would the perimeter of FDGH be?

4

<image> As shown in the figure, in △ABC, ∠ACB = 90°, D is a point on the extension line of BC, DF is perpendicular to AB at F, and DF intersects AC at E. How many right-angled triangles are there in the figure? Which angles are equal to ∠A? Which angles are complementary to ∠A? Please list them separately.

4

<image>As shown in the figure, add a condition: _____________ , to make △ADE ~ △ACB (write one condition only) __

∠ADE = ∠ACB

<image>FGH is geometrically an equilateral triangle. How would you calculate the perimeter of ECFHG?

17

<image>ACDE is a parallelogram. EDF is an isosceles triangle where the sides ED and DF are identical in length. Square FDGH includes a circle inscribed within. How long is FH in shape FDGH?

10

<image>ABC forms a sector. CBD takes on a sector shape. DBEF takes the form of a parallelogram. How long is HG in shape FEGH?

1

<image>Side CD shaping into an equilateral triangle beyond the rectangle. DEFG is a parallelogram. IJK is shaped as an equilateral triangle. What is the total area of the shape HFIKJ?

9*√(3)/4

<image>In the following groups, ______ are the shapes that are the same, and ______ are the shapes that are different.

(3)(5) (1)(2)(4)(6)

<image>As shown in the figure, the lower right corner of the quadrilateral paper $$ABCD$$ is folded inward to form $$\triangle PC′R$$, where $$∠B=120^{\circ}$$ and $$∠D=40^{\circ}$$, such that $$C′P \parallel AB$$ and $$RC′ \parallel AD$$. Then, $$∠C =$$ ______.

100°

<image> As shown in the figure, the diagonals AC and BD of trapezoid ABCD intersect at point O. If $ S_{△AOD} : S_{△ACD} = 1 : 4 $, then $ S_{△AOD} : S_{△BOC} = \_\_\_\_ $.

1:9

<image>(2014 Spring, Wuxi Final Exam) As shown in the figure, square ABCD with a side length of 3 cm is first translated 1 cm to the right and then 1 cm upward, resulting in square EFGH. The area of the shaded region is ____.

4

<image>Side DE converts into an equilateral triangle. Square EFGH encompasses an inscribed circle. What would the perimeter of inscribed circle be?

72*√(3)*π

<image>(2012 Autumn, Jiaxing School-level Midterm) As shown in the figure, CD is the diameter of circle O, and point C is on the minor arc subtended by chord AB. If point C is the midpoint of arc AB, then the measure of ∠AMD is ____.

90

<image>(2014 Autumn • Final Exam at Funan County) Xiaoming accidentally broke a triangular piece of glass into four pieces as shown in the diagram (labeled 1, 2, 3, and 4). In your opinion, which one of these pieces should be taken along to be able to match a piece of triangular glass of the original size? You should take piece ____, explain the reason.

2

<image>DCEF is geometrically a square. What is the measurement of DG in the shape DFG?

48*√(3)

<image>As shown in the figure, in the rectangular coordinate system, points $ A $ and $ C $ are on the $ x $-axis and $ y $-axis respectively. $ CB \parallel OA $, $ CB=8 $, $ OC=8 $, $ OA=16 $. (1) Directly write down the coordinates of points $ A $, $ B $, and $ C $ and the area of quadrilateral $ OABC $. (2) The moving point $ P $ starts from the origin $ O $ and moves towards $ A $. It stops when the line $ PC $ divides the quadrilateral $ OABC $ into two parts of equal area. Find the coordinates of point $ P $. (3) Under the condition of (2), is there a point $ Q $ on the $ y $-axis such that the area of triangle $ CPQ $ is equal to the area of quadrilateral $ OABC $? If such a point exists, find the coordinates of point $ Q $; if not, please explain why.

96

<image>As shown in the figure, in the rectangle $$ABCD$$, the diagonals $$AC$$ and $$BD$$ intersect at point $$O$$, $$AE \perp BD$$ at point $$E$$, and $$∠AOB = 50^{\circ}$$. What is the degree measure of $$∠BAE$$? ______ .

25

<image>(2014 Autumn • Chiping Middle School Midterm) As shown in the figure, point P is inside ∠AOB. Points M and N are the symmetric points of point P with respect to AO and BO, respectively. Points E and F are on the sides OA and OB, respectively. If the perimeter of triangle △PEF is 15, then the length of MN is ____.

15

<image>As shown in the figure: AB=AC, ∠A=50°, point O is a point inside △ABC, and ∠OBC=∠ACO, then ∠BOC=__ .__

115°

<image>ABCD conforms a square. CBE forms a sector. GFHI is geometrically a parallelogram. What would the perimeter of GFHI be?

6*√(3) + 14

<image>As shown in the picture, a street lamp is 8 meters above the ground, and Xiao Ming, who is 1.6 meters tall, is standing at point A, which is 20 meters from the base of the lamp (point O). The length of Xiao Ming's shadow is (___) meters.

5

<image> As shown in the figure, given that $\triangle ABC \cong \triangle BAE$, $\angle ABE = 60^{\circ}$, and $\angle E = 92^{\circ}$, then the measure of $\angle ABC$ is ______ degrees.

28

<image>As shown in the figure, given that ∠1 = 30°, then ∠2 = ____, ∠3 = ____.

150

<image>Side DE stretching into an equilateral triangle outside the rectangle. FDGH conforms a square. How much is the perimeter of FDGH?

208*√(3)/3

<image>FEG is a sector. What is the measurement of the perimeter of FEG?

45*π/2 + 90

<image>As shown in the figure, sunlight enters the classroom through the window, with the shadow of the window frame AB on the ground being DE=1.8m. The distance from the bottom edge of the window to the ground BC is 1m, and EC is 1.2m. What is the height of the window AB in meters?

1.5

<image> As shown in the figure, $$∠B=30^{\circ}$$. If $$AB \parallel CD$$ and $$CB$$ bisects $$∠ACD$$, then $$∠ACD =$$ ______ degrees.

60

<image>(2012 Spring • Hejiang County School Midterm) As shown in the , in △ABC, AE is the median, AD is the angle bisector, and AF is the altitude, then: (1) ∠BAC = 2____; (2) BC = 2____; (3) ____ = 90°.

\angle BAD

<image>As shown in the figure, in triangle ABC, ∠A=30°, ∠C=60°, DC=1cm, DE ⊥ bisects AB. Then AD=______cm.

1

<image> (2015 Autumn • Cangnan County Final) As shown in the figure, lines AB and CD intersect at point O, and OE bisects ∠BOD. If ∠AOE = 144°, then the degree measure of ∠AOC is ____.

72

<image> As shown in the figure, D is a point on side AB of △ABC. Through point D, draw DE ∥ BC, intersecting AC at point E. Through point E, draw EF ∥ AB, intersecting BC at point F. Then, the number of triangles similar to △ABC in the figure is ____. (excluding △ABC)

2

<image>As shown in the figure, the side length of square $$ABCD$$ is $$1$$. Points $$P$$ and $$Q$$ are on sides $$AB$$ and $$DA$$ respectively. When the perimeter of $$\triangle APQ$$ is $$2$$, find the measure of $$∠PCQ$$.

45^\circ

<image>ABC is defined as a sector. CBD is defined as a sector. What is the total length of the perimeter of DBE?

9*√(2) + 18

<image>As shown in the , line AB and CD intersect at point O. If ∠1 + ∠2 = 100°, then ∠BOC is equal to ______.

130

<image>If the length of the AB side is 20, the length of the AD side is $2x + 5$, the length of the AC side is $x + 3.92$, the degree of the CAD angle is 25 and the degree of the CDA angle is 30, compute the degree of the CBA angle. Round computations to 2 decimal places and round the value of the variable "x" to the nearest natural number.

23.58

<image>ABCD takes the form of a parallelogram. DCE takes the form of a sector. What is the measurement of EF in the shape ECF?

21*√(2)

<image>(Autumn 2015, Midterm Examination in Binhu District) As shown in the figure, in quadrilateral ABCD, AD ∥ BC, ∠B = 90°, AB = AD = 4 cm, and BC = 7 cm. Now, a paper in the shape of quadrilateral ABCD is to be cut to obtain an isosceles triangle with leg length of 3 cm (Requirements: One vertex of the isosceles triangle coincides with one vertex of quadrilateral ABCD, and the other two vertices lie on the sides of quadrilateral ABCD). Then, there are ____ possible values for the length of the base of the cut isosceles triangle.

4

<image>ABC is an isosceles triangle with AB and BC being equal in length. CBDE conforms a square. EDFG is a parallelogram. Please provide the length of GF in shape EDFG.

7

<image>Point P is a moving point on the ellipse =1 with its major axis on the x-axis. F_{1} and F_{2} are the two foci of the ellipse, and the semi-focal distance is c. Then, the maximum value of |PF_{1}|•|PF_{2}| is (__)

a^2

<image>Side CD is designed as an equilateral triangle. ECF takes on a sector shape. GHI defines an equilateral triangle. Square HIJK has an inner inscribed circle. What is the overall perimeter of circle?

3*π

<image>Side CD expanding into an equilateral triangle beyond the rectangle. DEFG conforms a square. GFHI is geometrically a square. Could you specify the length of HF in shape GFHI?

18

<image>As shown in the figure, in right triangle ABC, ∠C=90°, AD bisects ∠BAC and intersects BC at D, with BD:DC=5:4 and BC=18 cm. Then the distance from D to side AB is ___ cm.

8

<image>As shown in the figure, a right-angled triangular piece of paper is folded so that points B and C overlap, with the crease being DE. Connect DC. If AC = 6 cm, ∠ACB = 90°, ∠B = 30°, then the perimeter of △ADC is ______cm.

18

<image>As shown in the figure, points C and F are on BE, ∠1 = ∠2, and BC = EF. Please add one more condition: ___ (just one), to make △ABC ≌ △DEF.

∠B = ∠E

<image>ABCD takes the form of a parallelogram. How much is the perimeter of ECFG?

72 + 54*√(3)

<image>As shown in the figure, a rectangular piece of paper is folded along EF, causing points D and C to fall on positions $ D_{1} $ and $ C_{1} $ respectively. If ∠EFB=65°, then ∠BFC_{1}=__▲__°.

50

<image>Explore: (1) As shown in Figure ①, what is the relationship between ∠1 + ∠2 and ∠B + ∠C? Why? (2) Folding triangle △ABC along DE in Figure ①, we get Figure ②. Fill in the blank: ∠1 + ∠2 ______ ∠B + ∠C (fill in with “>”, “<”, or “=”), and when ∠A = 40°, ∠B + ∠C + ∠1 + ∠2 = ______; (3) As shown in Figure ③, which is obtained by folding △ABC along DE in Figure ①, if ∠A = 30°, then x + y = 360° - (∠B + ∠C + ∠1 + ∠2) = 360° - ______ = ______. Guess the relationship between ∠BDA + ∠CEA and ∠A: ______.

=

<image> As shown in the figure, $$\triangle ABC$$ ≌ $$\triangle DEF$$, $$∠A = 25^{\circ}$$, $$∠B = 65^{\circ}$$, $$BF = 3cm$$. Find the measure of $$∠DFE$$ and the length of $$EC$$.

90°

<image>In the figure, in $$\triangle ABC$$, $$∠ACB=90^{\circ}$$, $$D$$, $$E$$, and $$F$$ are the midpoints of sides $$AB$$, $$AC$$, and $$BC$$ respectively. Given that $$CD=3$$, find the length of $$EF$$.

3

<image>According to the figure, square ABCD with a side length of 3 cm is first translated 1 cm to the right and then 1 cm upwards to obtain square EFGH. The area of the shaded region is _________ square centimeters.

4

<image>ABCD is a square. ECF is an isosceles triangle in which EC equals CF in length. FCG takes the form of a sector. In shape FCG, what is the length of GC?

9

<image>DCEF is geometrically a square. What is the overall area measurement of the shape DFGH?

4*√(2)