Sicong/RL_toy_math_4 · Datasets at Hugging Face

images images listlengths 1 1	problem stringlengths 7 3.71k	answer stringlengths 1 34
	<image>The tailor folds a rectangular piece of cloth ABCD along AE, making point D fall on point F on side BC. If ∠BAF = 50°, then ∠DAE = ______°.	20
	<image>ABC is identified as a sector. CBDE forms a square. Square FDGH holds an inscribed circle. What is the total boundary length of circle?	3√(3)π
	<image>In triangle $$\triangle ABC$$, $$DE \parallel BC$$, $$DF \parallel AC$$, $$AE = 3$$, $$AC = 5$$, $$BC = 10$$. Find the length of $$CF$$.	6
	<image> As shown in the figure, DE ∥ GF ∥ BC, and AB ∥ EF ∥ DC. (1) What is the relationship between ∠B and ∠E? Why? (2) What is the relationship between ∠B and ∠F? Why?	∠E+∠B=180°
	<image>DCE is an isosceles triangle, having DC and CE of the same length. In shape ECFG, how long is GF?	1
	<image>ABCD is identified as a parallelogram. DCE is defined as a sector. ECF is an isosceles triangle, with EC equalling CF. Arc GH takes the shape of a semi-circle. Can you tell me the length of GH in shape FCGH?	3*π/2
	<image>EDF is an isosceles triangle where the lengths of ED and DF are equal. Can you specify the length of the base EF in the isosceles triangle EDF?	44*√(3)
	<image>ABCD forms a square. ADEF is identified as a parallelogram. FEGH is geometrically a square. Square GEIJ holds an inscribed circle. How much is the area of the entire circle?	9801*π/4
	<image>ADEF is a square. AFGH is geometrically a parallelogram. What is the total boundary length of AFGH?	150 + 64*√(6)
	<image>ABC is an isosceles triangle where AB and BC are of equal length. CBDE is geometrically a square. Side FG becoming an equilateral triangle outside the rectangle. How do you calculate the area of the shape DBFHG?	√(3)/4 + 3/2
	<image>(2015 Spring • Hubei School Level Midterm) As shown in the figure, lines AB and CD intersect at point O. EO is perpendicular to AB with the foot of the perpendicular being O. If ∠EOC = 35°, then ∠AOD = ____ degrees.	125
	<image>ACDE is a square. AEF is an isosceles triangle where the length of AE equals the length of EF. Side GH takes the shape of an equilateral triangle inside the rectangle. What's the perimeter of FEGIH?	256
	<image>As shown in the figure, \textit{AB} ∥ \textit{CD}, the line \textit{EF} intersects \textit{AB} and \textit{CD} at points \textit{E} and \textit{F} respectively. \textit{EG} bisects ∠\textit{AEF}. If ∠1 = 40°, then the degree measure of ∠2 is ___.	100
	<image>If the area of the ABCD parallelogram is 114 and the area of the EAB sector is 189.97, compute the degree of the BAD angle. Assume $\pi=3.14$. Round computations to 2 decimal places.	27.39
	<image>CDE is geometrically an equilateral triangle. Square HGIJ contains a circle that is inscribed. How do you find the perimeter of circle?	2√(3)π
	<image> (2013 Spring•Chaoyang District School-level Final Exam) Given: as shown in the diagram, AB and CD are straight lines, DF intersects AB at point E, and EG intersects CD at point O. If ∠BEF = 124°, ∠D = 56°, and ∠DEO = 60°, then the degree measure of ∠COE is ____.	116
	<image>As shown in the figure, in the parallelogram $$ABCD$$, extend $$DC$$ to $$F$$, connect $$AF$$, intersect $$BC$$ at point $$G$$, intersect $$BD$$ at point $$E$$. There are ____ pairs of similar triangles in the figure.	6
	<image>ACD forms a sector. DEFG is geometrically a square. Could you specify the length of FE in shape DEFG?	12
	<image>Side CD emerges as an equilateral triangle. ECF is an isosceles triangle, and the sides EC and CF are of equal length. What is the total surface area of the shape FCG?	338*√(3)
	<image>ABCD conforms a square. CBE is geometrically a sector. Side FG becoming an equilateral triangle outside the rectangle. IJK is in the shape of an equilateral triangle. Please provide the length of JK in shape GHIKJ.	8
	<image>Side CD becomes an equilateral triangle. FEGH takes the form of a parallelogram. How would you calculate the total area of the shape FEGH?	1975*√(6)
	<image>Arc FG constitutes a semi-circle. What is the area size of the entire shape ECFG?	-9π/2 + 18√(2)
	<image>Side CD develops into an equilateral triangle. ECF is geometrically a sector. FCGH is a parallelogram. What is the overall perimeter of FCGH?	100
	<image>ABC is identified as a sector. Side DE developing into an equilateral triangle outside. Side GH develops into a semi-circle. What is the arc length GH in the shape FDGH?	21*π/2
	<image>Side EF materializes as an equilateral triangle. FGHI is identified as a parallelogram. IHJK conforms a parallelogram. What is the length of KJ in shape IHJK?	84
	<image> As shown in the figure, there are two overlapping right-angled triangles. By translating one of the right-angled triangles along the direction of BC, we obtain △DEF. If AB=8cm, BE=4cm, and DH=3cm, then the area of the shaded part in the figure is (___) cm^{2}.	26
	<image>Six squares (as shown in the figure) overlap, with the connection points exactly at the midpoints of the squares. The side length of each square is \( a \). Then the perimeter of the figure is \_.	14a
	<image>ABC is an isosceles triangle where AB is the same length as BC. CBDE is geometrically a parallelogram. What is the total surface area of the shape EDF?	√(3)/2
	<image>ABCD is geometrically a square. CBEF is geometrically a square. CFGH is a square. Side IJ shapes into an equilateral triangle inside the rectangle. How do you find the area of the entire shape HGIKJ?	7644 - 8281*√(3)/4
	<image>DCE is a sector. ECFG forms a square. GFH forms a sector. What is the area of the entire shape GFH?	1225*π/4
	<image>ABC takes on a sector shape. Side DE emerges as an equilateral triangle. EFG forms a sector. How would you calculate the perimeter of EFG?	π/4 + 2
	<image>DCE is an isosceles triangle where the lengths of DC and CE are equal. In shape ECF, how long is FC?	65
	<image>CBD is an isosceles triangle where CB = BD. EBF takes on a sector shape. Please provide the length of arc EF in sector EBF.	3*π/4
	<image>As shown in the figure, △ACB ≌ △A′CB′, and ∠BCB′=30°. Then, the measure of ∠ACA′ is ______°.	30
	<image>As shown in the figure, the area of the circle is equal to the area of the rectangle. The circumference of the circle is 12.56 centimeters. Find the perimeter of the rectangle.	16.56
	<image>The area of each small square in the figure is 4 square centimeters. Find the area of the shaded part.	3.44
	<image>ACD is an isosceles triangle where AC is the same length as CD. Side EF is designed as an equilateral triangle. What is the perimeter of FGHI?	12
	<image>As shown in the figure: The area of the trapezoid is 45 square meters, the height is 6 meters, and the area of triangle AED is 5 square meters. Find the area of the shaded region.	20
	<image>ABC is an isosceles triangle, with AB equalling BC. CBDE is a square. FGH forms an equilateral triangle. GHIJ forms a square. Can you tell me the area of the whole shape GHIJ?	1
	<image>If the area of the ABCD rectangle is 114, the ABEF shape is a combination of a rectangle and a semi-circle, the perimeter of the ABEF shape is 64, the length of the BH side is $5x - 39$, the length of the BE side is $5x - 39.45$, the degree of the HBE angle is 30 and the degree of the BHE angle is 70, compute the length of the AD side of the ABCD rectangle. Assume $\pi=3.14$. Round computations to 2 decimal places and round the value of the variable "x" to the nearest natural number.	6.18
	<image>DEF is shaped as an equilateral triangle. Side GH constructs an equilateral triangle. What is the total length of the perimeter of EFGIH?	100 + 84*√(3)
	<image>As shown in the figure, in △ABC, BC = 5cm, BP and CP are the angle bisectors of ∠ABC and ∠ACB respectively, and PD is parallel to AB, PE is parallel to AC. Find the perimeter of △PDE.	5
	<image>As shown in the figure, $$∠A=∠DBC$$, $$AB=3$$, $$AC=5$$, $$BC=4$$, $$DB=4.8$$. Find $$CD=$$_______.	4
	<image>The quadrilateral $A B C D$ has right angles only in corners $A$ and $D$. The numbers in the diagram give the respective areas of the triangles in which they are located. How big is the area of $A B C D$?	45
	<image>ABCD is a square. DCE is identified as a sector. ECFG conforms a parallelogram. GFHI is a parallelogram. What is the length of IH in shape GFHI?	11
	<image>As shown in the , for a staircase with a height of 3 meters and a slope length of 5 meters that is to be covered with carpet, what is the minimum length of carpet required in meters? If the staircase is 2 meters wide and the carpet costs 30 yuan per square meter, how much will the carpet cost in total?	7
	<image>DCEF conforms a square. Side HI is extended to form a semi-circle within the rectangle. How long is HI in shape DGHI?	24√(2)π
	<image> Draw the lines of symmetry for the following shapes.	略
	<image>CBDE forms a square. GFHI is a parallelogram. What is the area size of the entire shape GFHI?	9/2
	<image> In the right triangle △ABC, BD bisects ∠ABC, and DE is perpendicular to AB at point E; (1) Which segments in the figure are equal? And which angles are equal? (2) If AB = 10, BC = 8, and AC = 6, find the lengths of BE and AE as well as the perimeter of △AED.	8
	<image>As shown in the diagram, in △ABC, ∠ACB = 90°, CD is the altitude, ∠A = 30°, BD = 3, then AD = _____ .__	9
	<image>As shown in the figure, ABCD is a quadrilateral with perpendicular diagonals, and OB=OD. Please add an appropriate condition ___ to make ABCD a rhombus (only one condition is needed).	OA=OC
	<image>ADEF forms a square. What's the area of the entire shape ADEF?	48
	<image>ABC is in the form of a sector. Side DE takes the shape of an equilateral triangle. Please provide the length of IH in shape FGHI.	48*√(3)
	<image>ABCD is identified as a parallelogram. ECFG is geometrically a square. What is the total area measurement of the shape ECFG?	1296
	<image> As shown in the figure, ∠B = 45°, ∠C = 72°, then the degree measure of ∠1 is ______.	117
	<image>As shown in the image, the sum of the angles ∠1, ∠2, ∠3, and ∠4 is ____________.	300
	<image>ABC takes on a sector shape. CBD forms a sector. EFG makes up an equilateral triangle. What is the overall perimeter of DBEGF?	120*√(2) + 288
	<image>A vegetable plastic greenhouse is 10 meters long and 2 meters high. The cross-section is a semicircle (as shown in the image). What is the maximum planting area of this greenhouse in square meters?	40
	<image>ABCD conforms a square. Side EF takes the shape of an equilateral triangle. Arc HI takes the shape of a semi-circle. How can we calculate the perimeter of FGHI?	36*π + 150
	<image>As shown in the figure, it is known that $$∠$$ \textit{$$COB$$}$$=2∠$$ \textit{$$AOC$$}, and \textit{$$OD$$} bisects $$∠$$ \textit{$$AOB$$}, and $$∠$$ \textit{$$COD$$}$$=20^{\circ}$$. Find the degree measure of $$∠$$ \textit{$$AOB$$}.	120°
	<image> As shown in the figure, $$E$$ is a point on the extension line of square $$ABCD$$, and $$CE=AC.$$ Then $$∠E=$$ ______ .	22.5
	<image>DCEF is geometrically a parallelogram. FEGH is geometrically a parallelogram. HGI is geometrically a sector. Can you tell me the length of arc HI in sector HGI?	13*π/6
	<image>As shown in the figure, it is known that $$∠AOM$$ and $$∠MOB$$ are complementary angles, and $$∠BOC=30^{\circ}$$, $$OM$$ bisects $$∠AOC$$, $$ON$$ bisects $$∠BOC$$. $$(1)$$ Find the measure of $$∠MON$$; $$(2)$$ If it is known that $$∠AOB=80^{\circ}$$ and other conditions remain unchanged, find the measure of $$∠MON$$; $$(3)$$ If it is known that $$∠BOC=60^{\circ}$$ and other conditions remain unchanged, find the measure of $$∠MON$$; $$(4)$$ From $$(1)$$, $$(2)$$, and $$(3)$$ can you observe any pattern?	45°
	<image>The area of the shaded part in the figure is 40 square meters. Find the area of the annulus.	125.6
	<image>ABCD conforms a square. ADE is an isosceles triangle where AD is the same length as DE. Side FG stretching into a semi-circle outside. How do you calculate the area of the shape EDFG?	9*π/2 + 45
	<image>As shown in the figure, the length of the rectangle is 6 cm, and the width is 4 cm. Therefore, the area of the circle in the figure is ____ cm².	12.56
	<image>Side CD becoming an equilateral triangle outside the rectangle. DEF takes on a sector shape. Could you specify the length of HI in shape HGI?	2*√(3)/3
	<image>As shown in the figure: In △ABC, ∠B = ∠C, E is a point on AC, ED⊥BC, DF⊥AB, with the feet of the perpendiculars being D and F respectively. If ∠AED = 140°, then ∠C = ____ degrees, ∠A = ____ degrees, ∠BDF = ____ degrees.	50
	<image>As shown in the figure, \(l_{1} \parallel l_{2}\), then \(\angle 1 = ___\) degrees.	100
	<image>As shown in the figure, a rectangular strip of paper ABCD is folded along AF. Given that ∠ADB = 25°, what should the angle ∠BAF be to make AE parallel to BD?	57.5°
	<image>ABCD is a parallelogram. DCEF forms a square. Arc GH describes a semi-circle. Can you tell me the length of GH in shape ECGH?	π/2
	<image>What is the area of the shaded part in the following figure in square centimeters? (Unit: cm) (Note: The white part is a square)	39
	<image>ABCD is geometrically a square. FEGH is a parallelogram. HGIJ takes the form of a parallelogram. What is the length of JI in shape HGIJ?	52
	<image> As shown in the figure, there is a "August 1st" Army Day commemorative medal, and its outer contour is a regular pentagon. The size of angle ∠1 in the figure is ____°.	108°
	<image> As shown in the figure, triangle △ABC is rotated counterclockwise by 40° around point A to the position of △AED, making DC∥AB. The measure of ∠CAE is ______.	30
	<image>DCEF is geometrically a square. Side GH leading into an equilateral triangle beyond the rectangle. In the shape IGJK, what is the measurement of KJ?	90
	<image>ACD is an isosceles triangle with AC having the same length as CD. Square DCEF has a circle inscribed. Please provide the length of EC in shape DCEF.	2*√(3)/3
	<image>Calculate the area of the shaded region in the image below (unit: centimeters).	18.84
	<image>As shown in the figure, the rhombus ABCD with a side length of 2 cm is folded along the line l where side AB lies to form the quadrilateral ABEF. If ∠DAB=30°, then the area of the quadrilateral CDFE is (__) _	4
	<image>ABC is an isosceles triangle where AB is the same length as BC. CBD is an isosceles triangle, and the sides CB and BD are of equal length. Side EF develops into an equilateral triangle inside the rectangle. How can we calculate the perimeter of DBEGF?	11
	<image>Side DE creating an equilateral triangle. GHI is geometrically an equilateral triangle. IGJ is an isosceles triangle where IG and GJ are of equal length. Please provide the length of JG in shape IGJ.	40*√(3)
	<image> As shown in the figure, $$\triangle ABD$$ ≌ $$\triangle EBC$$, $$AB = 3 \text{cm}$$, $$AC = 8 \text{cm}$$. Then, $$DE =$$ ______ $$\text{cm}$$.	2
	<image>ABCD is a square. ADE is an isosceles triangle where the sides AD and DE are identical in length. EDFG is identified as a parallelogram. What is the total length of the perimeter of EDFG?	20
	<image>ABC is an isosceles triangle in which AB and BC are equal. DEF is classified as an equilateral triangle. What is the total perimeter measurement of EFGH?	15
	<image>CBD is an isosceles triangle in which CB and BD are equal. DBEF conforms a square. Arc GH is a semi-circle. How would you determine the perimeter of FEGH?	11√(3)(π + 10)/2
	<image> As shown in the figure, given that the radius of circle O is 4, ∠A = 45°, if the lateral surface of a cone's development diagram can completely overlap with sector OBC, then the radius of the base circle of this cone is _________.	4
	<image>Side DE is designed as an equilateral triangle. How do you find the perimeter of FDGH?	178 + 162*√(2)
	<image>If a pair of triangles is superimposed together as shown in the , then ∠AOB=____ degrees.	15
	<image>ABCD is geometrically a square. CBEF is a parallelogram. FEG is geometrically a sector. What's the area of the entire shape GEH?	1152*√(3)
	<image> As shown in the figure, in the quadrilateral ABCD, AD ∥ BC, BC > AD, and ∠B and ∠C are supplementary. When AB and CD are translated to the positions of EF and EG respectively, then △EFG is a (___) triangle. If AD = 2 cm and BC = 8 cm, then FG = (___).	right
	<image>CDE is an equilateral triangle. Could you specify the length of HG in shape DFGH?	4*√(3)
	<image>EDFG forms a square. GFH takes the form of a sector. How much area does the shape GFH cover?	38416*π/3
	<image>CBD forms a sector. DBEF forms a square. What is the entire perimeter length of DBEF?	308*√(3)
	<image>ABCD is geometrically a square. ADE is an isosceles triangle where the length of AD equals the length of DE. Side FG extends and forms a semi-circle. How can we calculate the perimeter of EDFG?	15*π + 108
	<image>(10 points) As shown in the figure, point O lies on the line AB, and OC is the bisector of ∠AOB. On the opposite side of line AB, ∠DOE=90°, with vertex at point O. (1) If ∠AOE=46°, what is the measure of ∠DOB? Please indicate the quantitative relationship between ∠AOE and ∠DOB; (2) Please indicate the quantitative relationship between ∠DOB and ∠COE, and explain the reason.	44°
	<image>Side CD constructs an equilateral triangle. ECF is in the form of a sector. FCGH forms a square. Side IJ builds into an equilateral triangle. How would you determine the perimeter of GCIKJ?	6
	<image>As shown in the figure, rotate the right triangle ABC around the right-angle vertex C by 90 degrees clockwise to obtain △A′B′C. Connect AA′. If ∠1 = 35°, then the measure of ∠B is (___).	80
	<image>ABCD is geometrically a square. DCEF takes the form of a parallelogram. FEGH takes the form of a parallelogram. HGIJ forms a square. In the shape HGIJ, what is the measurement of HJ?	1
	<image>CDE forms a triangle with all sides equal. GFHI is geometrically a square. IHJK is a parallelogram. In shape IHJK, how long is KJ?	10

<image>The tailor folds a rectangular piece of cloth ABCD along AE, making point D fall on point F on side BC. If ∠BAF = 50°, then ∠DAE = ______°.

20

<image>ABC is identified as a sector. CBDE forms a square. Square FDGH holds an inscribed circle. What is the total boundary length of circle?

3*√(3)*π

<image>In triangle $$\triangle ABC$$, $$DE \parallel BC$$, $$DF \parallel AC$$, $$AE = 3$$, $$AC = 5$$, $$BC = 10$$. Find the length of $$CF$$.

6

<image> As shown in the figure, DE ∥ GF ∥ BC, and AB ∥ EF ∥ DC. (1) What is the relationship between ∠B and ∠E? Why? (2) What is the relationship between ∠B and ∠F? Why?

∠E+∠B=180°

<image>DCE is an isosceles triangle, having DC and CE of the same length. In shape ECFG, how long is GF?

1

<image>ABCD is identified as a parallelogram. DCE is defined as a sector. ECF is an isosceles triangle, with EC equalling CF. Arc GH takes the shape of a semi-circle. Can you tell me the length of GH in shape FCGH?

3*π/2

<image>EDF is an isosceles triangle where the lengths of ED and DF are equal. Can you specify the length of the base EF in the isosceles triangle EDF?

44*√(3)

<image>ABCD forms a square. ADEF is identified as a parallelogram. FEGH is geometrically a square. Square GEIJ holds an inscribed circle. How much is the area of the entire circle?

9801*π/4

<image>ADEF is a square. AFGH is geometrically a parallelogram. What is the total boundary length of AFGH?

150 + 64*√(6)

<image>ABC is an isosceles triangle where AB and BC are of equal length. CBDE is geometrically a square. Side FG becoming an equilateral triangle outside the rectangle. How do you calculate the area of the shape DBFHG?

√(3)/4 + 3/2

<image>(2015 Spring • Hubei School Level Midterm) As shown in the figure, lines AB and CD intersect at point O. EO is perpendicular to AB with the foot of the perpendicular being O. If ∠EOC = 35°, then ∠AOD = ____ degrees.

125

<image>ACDE is a square. AEF is an isosceles triangle where the length of AE equals the length of EF. Side GH takes the shape of an equilateral triangle inside the rectangle. What's the perimeter of FEGIH?

256

<image>As shown in the figure, \textit{AB} ∥ \textit{CD}, the line \textit{EF} intersects \textit{AB} and \textit{CD} at points \textit{E} and \textit{F} respectively. \textit{EG} bisects ∠\textit{AEF}. If ∠1 = 40°, then the degree measure of ∠2 is ___.

100

<image>If the area of the ABCD parallelogram is 114 and the area of the EAB sector is 189.97, compute the degree of the BAD angle. Assume $\pi=3.14$. Round computations to 2 decimal places.

27.39

<image>CDE is geometrically an equilateral triangle. Square HGIJ contains a circle that is inscribed. How do you find the perimeter of circle?

2*√(3)*π

<image> (2013 Spring•Chaoyang District School-level Final Exam) Given: as shown in the diagram, AB and CD are straight lines, DF intersects AB at point E, and EG intersects CD at point O. If ∠BEF = 124°, ∠D = 56°, and ∠DEO = 60°, then the degree measure of ∠COE is ____.

116

<image>As shown in the figure, in the parallelogram $$ABCD$$, extend $$DC$$ to $$F$$, connect $$AF$$, intersect $$BC$$ at point $$G$$, intersect $$BD$$ at point $$E$$. There are ____ pairs of similar triangles in the figure.

6

<image>ACD forms a sector. DEFG is geometrically a square. Could you specify the length of FE in shape DEFG?

12

<image>Side CD emerges as an equilateral triangle. ECF is an isosceles triangle, and the sides EC and CF are of equal length. What is the total surface area of the shape FCG?

338*√(3)

<image>ABCD conforms a square. CBE is geometrically a sector. Side FG becoming an equilateral triangle outside the rectangle. IJK is in the shape of an equilateral triangle. Please provide the length of JK in shape GHIKJ.

8

<image>Side CD becomes an equilateral triangle. FEGH takes the form of a parallelogram. How would you calculate the total area of the shape FEGH?

1975*√(6)

<image>Arc FG constitutes a semi-circle. What is the area size of the entire shape ECFG?

-9*π/2 + 18*√(2)

<image>Side CD develops into an equilateral triangle. ECF is geometrically a sector. FCGH is a parallelogram. What is the overall perimeter of FCGH?

100

<image>ABC is identified as a sector. Side DE developing into an equilateral triangle outside. Side GH develops into a semi-circle. What is the arc length GH in the shape FDGH?

21*π/2

<image>Side EF materializes as an equilateral triangle. FGHI is identified as a parallelogram. IHJK conforms a parallelogram. What is the length of KJ in shape IHJK?

84

<image> As shown in the figure, there are two overlapping right-angled triangles. By translating one of the right-angled triangles along the direction of BC, we obtain △DEF. If AB=8cm, BE=4cm, and DH=3cm, then the area of the shaded part in the figure is (___) cm^{2}.

26

<image>Six squares (as shown in the figure) overlap, with the connection points exactly at the midpoints of the squares. The side length of each square is $ a $. Then the perimeter of the figure is \_.

14a

<image>ABC is an isosceles triangle where AB is the same length as BC. CBDE is geometrically a parallelogram. What is the total surface area of the shape EDF?

√(3)/2

<image>ABCD is geometrically a square. CBEF is geometrically a square. CFGH is a square. Side IJ shapes into an equilateral triangle inside the rectangle. How do you find the area of the entire shape HGIKJ?

7644 - 8281*√(3)/4

<image>DCE is a sector. ECFG forms a square. GFH forms a sector. What is the area of the entire shape GFH?

1225*π/4

<image>ABC takes on a sector shape. Side DE emerges as an equilateral triangle. EFG forms a sector. How would you calculate the perimeter of EFG?

π/4 + 2

<image>DCE is an isosceles triangle where the lengths of DC and CE are equal. In shape ECF, how long is FC?

65

<image>CBD is an isosceles triangle where CB = BD. EBF takes on a sector shape. Please provide the length of arc EF in sector EBF.

3*π/4

<image>As shown in the figure, △ACB ≌ △A′CB′, and ∠BCB′=30°. Then, the measure of ∠ACA′ is ______°.

30

<image>As shown in the figure, the area of the circle is equal to the area of the rectangle. The circumference of the circle is 12.56 centimeters. Find the perimeter of the rectangle.

16.56

<image>The area of each small square in the figure is 4 square centimeters. Find the area of the shaded part.

3.44

<image>ACD is an isosceles triangle where AC is the same length as CD. Side EF is designed as an equilateral triangle. What is the perimeter of FGHI?

12

<image>As shown in the figure: The area of the trapezoid is 45 square meters, the height is 6 meters, and the area of triangle AED is 5 square meters. Find the area of the shaded region.

20

<image>ABC is an isosceles triangle, with AB equalling BC. CBDE is a square. FGH forms an equilateral triangle. GHIJ forms a square. Can you tell me the area of the whole shape GHIJ?

1

<image>If the area of the ABCD rectangle is 114, the ABEF shape is a combination of a rectangle and a semi-circle, the perimeter of the ABEF shape is 64, the length of the BH side is $5x - 39$, the length of the BE side is $5x - 39.45$, the degree of the HBE angle is 30 and the degree of the BHE angle is 70, compute the length of the AD side of the ABCD rectangle. Assume $\pi=3.14$. Round computations to 2 decimal places and round the value of the variable "x" to the nearest natural number.

6.18

<image>DEF is shaped as an equilateral triangle. Side GH constructs an equilateral triangle. What is the total length of the perimeter of EFGIH?

100 + 84*√(3)

<image>As shown in the figure, in △ABC, BC = 5cm, BP and CP are the angle bisectors of ∠ABC and ∠ACB respectively, and PD is parallel to AB, PE is parallel to AC. Find the perimeter of △PDE.

5

<image>As shown in the figure, $$∠A=∠DBC$$, $$AB=3$$, $$AC=5$$, $$BC=4$$, $$DB=4.8$$. Find $$CD=$$_______.

4

<image>The quadrilateral $A B C D$ has right angles only in corners $A$ and $D$. The numbers in the diagram give the respective areas of the triangles in which they are located. How big is the area of $A B C D$?

45

<image>ABCD is a square. DCE is identified as a sector. ECFG conforms a parallelogram. GFHI is a parallelogram. What is the length of IH in shape GFHI?

11

<image>As shown in the , for a staircase with a height of 3 meters and a slope length of 5 meters that is to be covered with carpet, what is the minimum length of carpet required in meters? If the staircase is 2 meters wide and the carpet costs 30 yuan per square meter, how much will the carpet cost in total?

7

<image>DCEF conforms a square. Side HI is extended to form a semi-circle within the rectangle. How long is HI in shape DGHI?

24*√(2)*π

<image> Draw the lines of symmetry for the following shapes.

略

<image>CBDE forms a square. GFHI is a parallelogram. What is the area size of the entire shape GFHI?

9/2

<image> In the right triangle △ABC, BD bisects ∠ABC, and DE is perpendicular to AB at point E; (1) Which segments in the figure are equal? And which angles are equal? (2) If AB = 10, BC = 8, and AC = 6, find the lengths of BE and AE as well as the perimeter of △AED.

8

<image>As shown in the diagram, in △ABC, ∠ACB = 90°, CD is the altitude, ∠A = 30°, BD = 3, then AD = _____ .__

9

<image>As shown in the figure, ABCD is a quadrilateral with perpendicular diagonals, and OB=OD. Please add an appropriate condition ___ to make ABCD a rhombus (only one condition is needed).

OA=OC

<image>ADEF forms a square. What's the area of the entire shape ADEF?

48

<image>ABC is in the form of a sector. Side DE takes the shape of an equilateral triangle. Please provide the length of IH in shape FGHI.

48*√(3)

<image>ABCD is identified as a parallelogram. ECFG is geometrically a square. What is the total area measurement of the shape ECFG?

1296

<image> As shown in the figure, ∠B = 45°, ∠C = 72°, then the degree measure of ∠1 is ______.

117

<image>As shown in the image, the sum of the angles ∠1, ∠2, ∠3, and ∠4 is ____________.

300

<image>ABC takes on a sector shape. CBD forms a sector. EFG makes up an equilateral triangle. What is the overall perimeter of DBEGF?

120*√(2) + 288

<image>A vegetable plastic greenhouse is 10 meters long and 2 meters high. The cross-section is a semicircle (as shown in the image). What is the maximum planting area of this greenhouse in square meters?

40

<image>ABCD conforms a square. Side EF takes the shape of an equilateral triangle. Arc HI takes the shape of a semi-circle. How can we calculate the perimeter of FGHI?

36*π + 150

<image>As shown in the figure, it is known that $$∠$$ \textit{$$COB$$}$$=2∠$$ \textit{$$AOC$$}, and \textit{$$OD$$} bisects $$∠$$ \textit{$$AOB$$}, and $$∠$$ \textit{$$COD$$}$$=20^{\circ}$$. Find the degree measure of $$∠$$ \textit{$$AOB$$}.

120°

<image> As shown in the figure, $$E$$ is a point on the extension line of square $$ABCD$$, and $$CE=AC.$$ Then $$∠E=$$ ______ .

22.5

<image>DCEF is geometrically a parallelogram. FEGH is geometrically a parallelogram. HGI is geometrically a sector. Can you tell me the length of arc HI in sector HGI?

13*π/6

<image>As shown in the figure, it is known that $$∠AOM$$ and $$∠MOB$$ are complementary angles, and $$∠BOC=30^{\circ}$$, $$OM$$ bisects $$∠AOC$$, $$ON$$ bisects $$∠BOC$$. $$(1)$$ Find the measure of $$∠MON$$; $$(2)$$ If it is known that $$∠AOB=80^{\circ}$$ and other conditions remain unchanged, find the measure of $$∠MON$$; $$(3)$$ If it is known that $$∠BOC=60^{\circ}$$ and other conditions remain unchanged, find the measure of $$∠MON$$; $$(4)$$ From $$(1)$$, $$(2)$$, and $$(3)$$ can you observe any pattern?

45°

<image>The area of the shaded part in the figure is 40 square meters. Find the area of the annulus.

125.6

<image>ABCD conforms a square. ADE is an isosceles triangle where AD is the same length as DE. Side FG stretching into a semi-circle outside. How do you calculate the area of the shape EDFG?

9*π/2 + 45

<image>As shown in the figure, the length of the rectangle is 6 cm, and the width is 4 cm. Therefore, the area of the circle in the figure is ____ cm².

12.56

<image>Side CD becoming an equilateral triangle outside the rectangle. DEF takes on a sector shape. Could you specify the length of HI in shape HGI?

2*√(3)/3

<image>As shown in the figure: In △ABC, ∠B = ∠C, E is a point on AC, ED⊥BC, DF⊥AB, with the feet of the perpendiculars being D and F respectively. If ∠AED = 140°, then ∠C = ____ degrees, ∠A = ____ degrees, ∠BDF = ____ degrees.

50

<image>As shown in the figure, $l_{1} \parallel l_{2}$, then $\angle 1 = ___$ degrees.

100

<image>As shown in the figure, a rectangular strip of paper ABCD is folded along AF. Given that ∠ADB = 25°, what should the angle ∠BAF be to make AE parallel to BD?

57.5°

<image>ABCD is a parallelogram. DCEF forms a square. Arc GH describes a semi-circle. Can you tell me the length of GH in shape ECGH?

π/2

<image>What is the area of the shaded part in the following figure in square centimeters? (Unit: cm) (Note: The white part is a square)

39

<image>ABCD is geometrically a square. FEGH is a parallelogram. HGIJ takes the form of a parallelogram. What is the length of JI in shape HGIJ?

52

<image> As shown in the figure, there is a "August 1st" Army Day commemorative medal, and its outer contour is a regular pentagon. The size of angle ∠1 in the figure is ____°.

108°

<image> As shown in the figure, triangle △ABC is rotated counterclockwise by 40° around point A to the position of △AED, making DC∥AB. The measure of ∠CAE is ______.

30

<image>DCEF is geometrically a square. Side GH leading into an equilateral triangle beyond the rectangle. In the shape IGJK, what is the measurement of KJ?

90

<image>ACD is an isosceles triangle with AC having the same length as CD. Square DCEF has a circle inscribed. Please provide the length of EC in shape DCEF.

2*√(3)/3

<image>Calculate the area of the shaded region in the image below (unit: centimeters).

18.84

<image>As shown in the figure, the rhombus ABCD with a side length of 2 cm is folded along the line l where side AB lies to form the quadrilateral ABEF. If ∠DAB=30°, then the area of the quadrilateral CDFE is (__) _

4

<image>ABC is an isosceles triangle where AB is the same length as BC. CBD is an isosceles triangle, and the sides CB and BD are of equal length. Side EF develops into an equilateral triangle inside the rectangle. How can we calculate the perimeter of DBEGF?

11

<image>Side DE creating an equilateral triangle. GHI is geometrically an equilateral triangle. IGJ is an isosceles triangle where IG and GJ are of equal length. Please provide the length of JG in shape IGJ.

40*√(3)

<image> As shown in the figure, $$\triangle ABD$$ ≌ $$\triangle EBC$$, $$AB = 3 \text{cm}$$, $$AC = 8 \text{cm}$$. Then, $$DE =$$ ______ $$\text{cm}$$.

2

<image>ABCD is a square. ADE is an isosceles triangle where the sides AD and DE are identical in length. EDFG is identified as a parallelogram. What is the total length of the perimeter of EDFG?

20

<image>ABC is an isosceles triangle in which AB and BC are equal. DEF is classified as an equilateral triangle. What is the total perimeter measurement of EFGH?

15

<image>CBD is an isosceles triangle in which CB and BD are equal. DBEF conforms a square. Arc GH is a semi-circle. How would you determine the perimeter of FEGH?

11*√(3)*(π + 10)/2

<image> As shown in the figure, given that the radius of circle O is 4, ∠A = 45°, if the lateral surface of a cone's development diagram can completely overlap with sector OBC, then the radius of the base circle of this cone is _________.

4

<image>Side DE is designed as an equilateral triangle. How do you find the perimeter of FDGH?

178 + 162*√(2)

<image>If a pair of triangles is superimposed together as shown in the , then ∠AOB=____ degrees.

15

<image>ABCD is geometrically a square. CBEF is a parallelogram. FEG is geometrically a sector. What's the area of the entire shape GEH?

1152*√(3)

<image> As shown in the figure, in the quadrilateral ABCD, AD ∥ BC, BC > AD, and ∠B and ∠C are supplementary. When AB and CD are translated to the positions of EF and EG respectively, then △EFG is a (___) triangle. If AD = 2 cm and BC = 8 cm, then FG = (___).

right

<image>CDE is an equilateral triangle. Could you specify the length of HG in shape DFGH?

4*√(3)

<image>EDFG forms a square. GFH takes the form of a sector. How much area does the shape GFH cover?

38416*π/3

<image>CBD forms a sector. DBEF forms a square. What is the entire perimeter length of DBEF?

308*√(3)

<image>ABCD is geometrically a square. ADE is an isosceles triangle where the length of AD equals the length of DE. Side FG extends and forms a semi-circle. How can we calculate the perimeter of EDFG?

15*π + 108

<image>(10 points) As shown in the figure, point O lies on the line AB, and OC is the bisector of ∠AOB. On the opposite side of line AB, ∠DOE=90°, with vertex at point O. (1) If ∠AOE=46°, what is the measure of ∠DOB? Please indicate the quantitative relationship between ∠AOE and ∠DOB; (2) Please indicate the quantitative relationship between ∠DOB and ∠COE, and explain the reason.

44°

<image>Side CD constructs an equilateral triangle. ECF is in the form of a sector. FCGH forms a square. Side IJ builds into an equilateral triangle. How would you determine the perimeter of GCIKJ?

6

<image>As shown in the figure, rotate the right triangle ABC around the right-angle vertex C by 90 degrees clockwise to obtain △A′B′C. Connect AA′. If ∠1 = 35°, then the measure of ∠B is (___).

80

<image>ABCD is geometrically a square. DCEF takes the form of a parallelogram. FEGH takes the form of a parallelogram. HGIJ forms a square. In the shape HGIJ, what is the measurement of HJ?

1

<image>CDE forms a triangle with all sides equal. GFHI is geometrically a square. IHJK is a parallelogram. In shape IHJK, how long is KJ?

10