Sicong/RL_toy_math_4 · Datasets at Hugging Face

images images listlengths 1 1	problem stringlengths 7 3.71k	answer stringlengths 1 34
	<image>As shown in the figure, A and D are two points on circle O, and BC is the diameter. If ∠D = 35°, then the degree of ∠COA is ________.	70°
	<image>ABC is an isosceles triangle where the lengths of AB and BC are equal. CBD is in the form of a sector. What is the total area of shape DBE?	32*√(3)
	<image>As shown in the , AB is parallel to CD, DE is perpendicular to AC with the foot of the perpendicular being E, and ∠A is 55°. What is the degree measure of ∠D?	35°
	<image>As shown in the figure, △ABC is an equilateral triangle. △AP′B can coincide with △APC after rotation. So: (1) Identify the center of rotation; (2) Determine the degree of the rotation angle; (3) Find the degree of ∠PAP′.	60°
	<image>ABCD conforms a square. CBE takes on a sector shape. EBF is identified as a sector. FBG is an isosceles triangle in which FB and BG are equal. Could you specify the length of GB in shape FBG?	13
	<image> As shown in the figure, the circle is divided into several equal parts and rearranged into an approximate rectangle. The width of the rectangle is 4 cm, and the length is ( ___ ) cm.	12.56
	<image>Side CD transforming into an equilateral triangle beyond the rectangle. DEFG is identified as a parallelogram. IHJK is geometrically a square. In shape IHJK, how long is KJ?	3
	<image>ABCD takes the form of a parallelogram. DCEF is a square. FEG is a sector. GEH is an isosceles triangle where GE is the same length as EH. Could you specify the length of HE in shape GEH?	9
	<image>Side CD morphs into an equilateral triangle. ECF is identified as a sector. FCG is defined as a sector. Square GCHI has an inner inscribed circle. How much is the perimeter of circle?	36*π
	<image> Given: As shown in the figure, in the isosceles trapezoid ABCD, AD ∥ BC, AB = DC, and AC and BD intersect at point O. Prove that OB = OC.	OB=OC
	<image>As shown in the figure, it is known that line AB intersects with line CD at point O, OE bisects ∠AOC, ray OF is perpendicular to CD at point O, and ∠BOF = 32°. Find the degree measure of ∠COE.	61°
	<image>ACDE is a parallelogram. EDFG conforms a square. What is the total area of the shape EDFG?	18
	<image>(2014 Autumn • Mid-term in Hailing District) As shown in the figure, △ABO ≌ △CBO, if ∠A = 85°, ∠ABO = 35°, then the measure of ∠BOC is ____°.	60
	<image> As shown in the figure, quadrilateral $$ABCD$$ is a rhombus, $$O$$ is the intersection point of its two diagonals. Three lines passing through point $$O$$ divide the rhombus into shaded and unshaded parts. When the lengths of the two diagonals of the rhombus are $$6$$ and $$8$$ respectively, the area of the shaded part is ______.	12
	<image> As shown in the figure, given a rectangular paper AB, point E is the midpoint of AB, point G is a point on BC, and ∠BEG > 60°. Now fold the paper along the line EG so that point B falls at point H on the paper. Connect AH, then the number of angles equal to ∠BEG is (___).	3
	<image> The Olympic Park observation tower consists of five independent towers of varying heights arranged in a staggered manner. During a comprehensive practical activity class, a group of students decided to measure the height of the tallest tower using a clinometer (assuming the height of the clinometer is negligible). Their method is as follows: as shown in the figure, they first measure the angle of elevation of the top of the tallest tower \(A\) from point \(B\) to be \(45^{\circ}\). They then walk directly 90 meters towards the base of the tallest tower to reach point \(C\), where they again measure the angle of elevation of the top of the tallest tower \(A\), which is \(58^{\circ}\). Please help them calculate the approximate height \(AD\) of the tallest tower. (Reference data: \(\sin 58^{\circ} \approx 0.85\), \(\cos 58^{\circ} \approx 0.53\), \(\tan 58^{\circ} \approx 1.60\))	240
	<image>As shown in the figure, points E, F, G, and H are the midpoints of the sides AB, BC, CD, and DA of parallelogram ABCD, respectively. Prove that △BEF ≌ △DGH.	△BEF ≌ △DGH
	<image>As shown in the figure, point P is inside ∠AOB. Points \( P_1 \) and \( P_2 \) are the symmetric points of point P with respect to OA and OB, respectively. \( P_1 P_2 \) intersects OA at M and intersects OB at N. If \( P_1 P_2 = 5 \) cm, then the perimeter of △PMN is ______.	5cm
	<image>ABCD conforms a parallelogram. DCE is an isosceles triangle where DC and CE are of equal length. GFH takes on a sector shape. In sector GFH, what is the length of arc GH?	π/2
	<image>The intersection points of the line \textit{y}=\textit{kx}+\textit{b} with the coordinate axes are shown in the figure. When \textit{y}<0, the range of values for \textit{x} is (__ )	x < 2
	<image> As shown in the figure, there are two slides of equal length leaning against a wall. It is known that the height AC of the left slide is equal to the horizontal length DF of the right slide. The sum of the angles ∠ABC and ∠DFE formed by these two slides with the ground is ____ degrees.	90
	<image>【Problem】In order to increase the distance tourists walk to view the garden, a straight road with a constant width of \textit{a} meters in the parallelogram-shaped garden, as shown in Figure 1, was changed to a curved road with a constant width of \textit{a} meters, as shown in Figure 2. The remaining areas after the road reconstruction (i.e., the shaded areas in the figure) are denoted as \textit{S}_{1} and \textit{S}_{2}, respectively. Then, \textit{S}_{1} ________ \textit{S}_{2} (fill in with “>”, “=”, or “<”).	=
	<image>As shown in the , ∠BAC=110°, if MP and NQ are the perpendicular bisectors of AB and AC respectively, then ∠PAQ=____.	40
	<image>As shown in the figure, the main peak of a certain mountain has an elevation of approximately $$600$$ meters. A telecom signal transmission tower $$BC$$ is built on the main peak $$AB$$. Xiao Ming measures the angle of elevation to the peak $$B$$ from point $$P$$ at the foot of the mountain to be $$α$$, and the angle of elevation to the top of the transmission tower $$C$$ to be $$β$$. Given that $$\tan \alpha =\dfrac{3}{5}$$ and $$\tan \beta =\dfrac{5}{8}$$, find the height of the transmission tower $$BC$$.	25
	<image>As shown in the figure, AC and BD intersect at O, and AB = DC, AC = BD. Prove that OB = OC.	OB=OC
	<image> As shown in the figure, the right-angle vertices of a pair of set squares coincide. (1) Compare the sizes of ∠EOM and ∠FON, and explain the reason; (2) If ∠FOM = 60°, find the degree measure of ∠EON.	120°
	<image>ABC is an isosceles triangle with AB having the same length as BC. Side DE developing into an equilateral triangle outside. HGI takes the form of a sector. In sector HGI, how long is arc HI?	12*π
	<image>As shown in the figure, points \textit{A}, \textit{B}, and \textit{C} are on circle \textit{O}, \textit{AO} ∥ \textit{BC}, ∠\textit{AOB} = 50°. Then the measure of ∠\textit{OAC} is _________. ___________________________________________________________________________________	25
	<image>As shown in the figure, AB=DE, AC=DF, and points E and C are on line BF, with BE=CF. Prove that △ABC≌△DEF.	△ABC≌△DEF
	<image>DCEF takes the form of a parallelogram. Side GH transforming into an equilateral triangle beyond the rectangle. IGJK forms a square. Could you specify the length of JG in shape IGJK?	12
	<image>As shown in the figure, in triangle △BAC, AB = AC, AB + BC = 13, and the perpendicular bisector MN of side AB intersects AC at D. Find the perimeter of △BCD. ________ _	13
	<image>In circle ⊙O, the chord AB = 2 cm, and ∠ACB = 30°, then the diameter of ⊙O is ____ _______ __ cm.__	4
	<image>As shown in the figure, ⊙O is the circumcircle of triangle △ABC, and it is known that ∠ABO=50°. Then, the measure of ∠ACB is (______ )	40
	<image>DCE is an isosceles triangle in which DC and CE are equal. Square ECFG includes an inscribed circle. What is the entire perimeter length of inscribed circle?	π
	<image>As shown in the figure, given ∠O = 35°, and CD is the perpendicular bisector of OA, then the degree of ∠ACB is ______.	70°
	<image>The positions of real numbers \( a \) and \( b \) on the number line are shown in the figure. Then \( \|a\| \) ____ \( \|b\| \) (fill in with "\>", "\<" or "\=").	>
	<image>As shown in the , it is known that ∠COB = 2∠AOC, OD bisects ∠AOB, and ∠AOC = 40°. Find the measure of ∠BOD.	60
	<image>As shown in the , in the quadrilateral $$ABCD$$, $$∠1=∠2$$, $$∠D=72^{\circ}$$, then $$∠BCD=$$ ______ .	108°
	<image>As shown in the figure, point D is inside the isosceles right triangle ABC, with BC as the hypotenuse. If triangle ABD is rotated counterclockwise about point A to the position of triangle ACD', then ∠ADD' = ______.	45
	<image>As shown in the figure, AB = AC, AD ⊥ BC, and the foot of the perpendicular is D. If ∠BAD = 20°, then ∠BAC = ___ °.	40
	<image> As shown in the figure, $$\triangle ABC \cong \triangle DEF$$. Based on the information provided in the figure, write $$x =$$ ______ .	20
	<image>As shown in the figure, in an equilateral triangle, \( BC = 6 \text{ cm} \), the ray \( AG \parallel BC \), point \( E \) starts moving from point \( A \) along the ray \( AG \) at a speed of \( 1 \text{ cm/s} \). At the same time, point \( F \) starts moving from point \( B \) along the ray \( BC \) at a speed of \( 2 \text{ cm/s} \). Let the movement time be \( t \) \( (s) \). (1) Connect \( EF \). When \( EF \) passes through the midpoint of side \( AC \), find the value of \( t \). (2) During the movement, can the quadrilateral with vertices \( A \), \( C \), \( E \), and \( F \) be a rhombus? If yes, find the value of \( t \); if not, please explain why.	2
	<image>As shown in the , on a straight railway, points A and B are 25 km apart. Villages C and D are located such that DA=10 km, CB=15 km, DA is perpendicular to AB at point A, and CB is perpendicular to AB at point B. Now, a transfer station E needs to be constructed on AB such that the distances from villages C and D to station E are equal. How far should station E be built from point A?	15
	<image>ABCD takes the form of a parallelogram. DCE forms a sector. ECF is a sector. Can you tell me the length of FC in shape ECF?	4
	<image> As shown in the figure, AD is the median of △ABC, ∠ADC=60°. Fold △ADC along the line AD, making point C fall to C'. If BC' = 5, then BC = ____.	10
	<image>As shown in the figure, given that ∠1 = 70º and CD∥BE, what is the degree measure of ∠B? (__)	110
	<image>As shown in the figure, there are several types of cards A (square), B (square), and C (rectangle). If you want to form a large rectangle with a length of (a+2b) and a width of (a+b), you will need ______ of C type cards.	3
	<image>As shown in the figure, the area of the shaded region is ____ square centimeters.	4
	<image>ABCD conforms a square. CBEF is geometrically a square. CFG is an isosceles triangle, having CF and FG of the same length. In the isosceles triangle CFG, what are the measures of angle FCG and angle CGF?	67.5
	<image> As shown in the diagram, in △ABC, AB = AC, and CD bisects ∠ACB. Given that ∠A = 36°, then the measure of ∠BDC is ____.	72
	<image>ABC is an isosceles triangle where AB = BC. CBDE is a parallelogram. EDF is an isosceles triangle where the length of ED equals the length of DF. Side GH curves into a semi-circle. What is the arc length GH in the shape FDGH?	π
	<image>As shown in the diagram, a point A on the circumference of a coin coincides with the origin O of the number line. The coin rolls one complete turn along the positive direction of the number line, and point A coincides exactly with a point A' on the number line. If the radius of the coin is 1 unit length, what number does the point A' on the number line represent?	2\pi
	<image> As shown in the figure, the coordinates of $$A$$ and $$B$$ are $$(1,0)$$ and $$(0,2)$$, respectively. If the line segment $$AB$$ is translated to $$A_{1}B_{1}$$, then the value of $$a-b$$ is ______ .	0
	<image>As shown in the figure, in the parallelogram ABCD, point E is on side BC, and CE:BC=2:3. AC intersects DE at point F. If the area of △AFD is 9, then the area of △EFC is _______ .__	4
	<image>If the length of the AD side is 9, the length of the BE side is 22 and the degree of the BEA angle is 55, compute the perimeter of the ABCD parallelogram. Round computations to 2 decimal places.	54.08
	<image> (Spring 2011, Guangzhou school-level midterm) As shown in the figure, if ∠1 = 25°, then ∠2 = ____, ∠3 = ____, ∠4 = ____.	155°
	<image>In Figure ①, there is a rectangular piece of paper with a length of \(2a\) and a width of \(2b\). The area of the rectangle is clearly \(4ab\). Now, this rectangular piece of paper is cut along the dotted line in the figure, dividing it into 4 smaller rectangles, and then reassembled into a square as shown in Figure ②. (1) In Figure ②, the side length of the shaded square \(EFGH\) is: ________________; (2) Observing Figure ②, which shape's area is represented by the algebraic expression \((a - b)^{2}\)? How about the algebraic expression \((a + b)^{2}\)? (3) Express the area of the shaded square \(EFGH\) in Figure ② using two different methods, and write out the equal relationship between the algebraic expressions \((a + b)^{2}\), \((a - b)^{2}\), and \(4ab\); (4) Based on the equal relationship in question (3), solve the following problem: If \(a + b = 7\) and \(ab = 5\), find the value of \((a - b)^{2}\).	a-b
	<image> In the parallelogram $$ABCD$$, fold $$\triangle BCD$$ along $$BD$$ so that point $$C$$ falls at point $$E$$, and $$BE$$ intersects $$AD$$ at point $$O$$. Prove that $$OA=OE$$.	OA=OE
	<image>As shown in the , OC is a ray inside the angle ∠AOB. Triangle ODE is a triangle with a 60° angle, with its vertex at point O coinciding with the 60°, and side OD is on the ray that exactly bisects ∠AOC, while side OE is on the ray that exactly bisects ∠BOC. Find the degree measure of ∠AOB.	120
	<image>As shown in the figure, in the parallelogram $$ABCD$$, $$AB{=}5{,}AD{=}8{,}{DE}$$ bisects $$∠ADC$$. Then $$BE{=}$$______.	3
	<image>If the ADEF shape is a combination of a rectangle and a semi-circle and the area of the ADEF shape is 90, compute the perimeter of the ABCD square. Assume $\pi=3.14$. Round computations to 2 decimal places.	50.6
	<image>As shown in the figure, there is a square inside a rectangle. Given that AB = 7 cm and CD = 5 cm, find the perimeter of the large rectangle.	24
	<image> As shown in the figure, AD and AC are the diameter and chord of circle O respectively, and ∠CAD=30°. OB is perpendicular to AD and intersects AC at point B. If OB=5, then the length of BC is (___ ).	5
	<image>DCE is an isosceles triangle with DC and CE being equal in length. ECF is an isosceles triangle, with EC equalling CF. How long is the base EF in the isosceles triangle ECF?	39
	<image>DCEF is identified as a parallelogram. FEGH conforms a square. Square HGIJ contains an inscribed circle. What is the area of the entire circle?	1089*π
	<image>ABCD is geometrically a square. ADEF is geometrically a square. EDG is in the form of a sector. In shape EDG, what is the length of GD?	24
	<image>If the arc length of the blue sector is 17.99, the pink shape is a rectangle where a semi-circle has been removed from one side of it, the perimeter of the pink shape is 44 and the area of the brown square is 64, compute the degree of the angle marked with question mark. Assume $\pi=3.14$. Round computations to 2 decimal places.	86.4
	<image>(2010 Autumn • Yanping District Final Exam) As shown in the figure, in the parallelogram ABCD, AC and BD intersect at point O. Given that AB = 8, BC = 6, and the perimeter of △AOB is 18, what is the perimeter of △AOD? _____.	16
	<image>As shown in the figure, line AB and line CD intersect at point O. Ray OM bisects ∠AOC, and ON is perpendicular to OM. If ∠AOM = 35°, then the degree measure of ∠CON is (___)	55
	<image>As shown in the figure, a right triangle board ABC with a 30° angle has its right-angled vertex A on the line DE, and BC∥DE. Then ∠CAE is equal to (__)	30
	<image>As shown in the figure, to measure the height of the flagpole \( AB \), a rope hanging down from the top of the flagpole can be used. When the rope is vertical to the ground, it is found to be \( 1m \) longer than the height of the flagpole. When the rope is stretched to point \( C \) on the ground, the length of \( CB \) is measured to be \( 5m \). Determine the height of the flagpole \( AB \).	12
	<image>If the area of the ABCD parallelogram is 54, the ABEF shape is a rectangle where a semi-circle has been removed from one side of it, the length of the BE side is 9, the area of the ABEF shape is 96, the ADHIJ shape is a rectangle where an equilateral triangle has been removed from one side of it, the length of the DH side is 6 and the perimeter of the ADHIJ shape is 42, compute the degree of the BAD angle. Assume $\pi=3.14$. Round computations to 2 decimal places.	18.66
	<image>ABC takes the form of a sector. EDFG conforms a square. GFH is an isosceles triangle in which GF and FH are equal. In the isosceles triangle GFH, what are the measures of angle FGH and angle GHF?	45
	<image>(2010 Autumn • Shangrao County School Level Midterm) As shown in the figure, △ABC ≌ △ADE, ∠1 = 30°, then ∠BAD = ____.	30°
	<image>As shown in the figure, ∠A is an inscribed angle of circle O, and ∠A = 60°. Then the measure of ∠OBC is ____ degrees.	30
	<image>ABCD conforms a square. ADE is an isosceles triangle, having AD and DE of the same length. FGH is an equilateral triangle. In the shape HFI, what is the measurement of HI?	√(2)
	<image>Side CD transforming into an equilateral triangle beyond the rectangle. DEFG is geometrically a square. FEHI is geometrically a square. What is the measurement of the perimeter of FEHI?	20
	<image> As shown in the figure, the area of triangle ABE is 48 square decimeters, and BC = CD = DE. Then the area of the shaded part is ____ square decimeters.	16
	<image>Please complete the following proof process. Given: As shown in the figure, $$DE/\!/BC$$, $$BE$$ bisects $$∠ABC$$. Prove: $$∠1=∠3$$. ___ Proof: Because $$BE$$ bisects $$∠ABC$$ (Given), ___ therefore $$∠1=$$_______$$($$_ ________ ______ $$). ___ Also, because $$DE/\!/BC$$ (Given), ___ therefore $$∠2=$$_______$$($$_ ________________ $$). ___ Therefore, $$∠1=∠3$$ ($$ _________________ $$).	∠1=∠3
	<image>As shown in the figure, in △ABC, AB = AC, and the altitude AD is drawn on the base BC. Prove: ∠B = ∠C	∠B=∠C
	<image>As shown in the figure, in $$\triangle ABC$$, $$BC \perp AC$$, $$CD$$ is the height on side $$AB$$, if $$AB=10cm$$, $$BC=6cm$$, $$AC=8cm$$, then $$CD=$$________	4.8
	<image> As shown in the figure, in triangle △ABC, AD is the angle bisector. If ∠B = ∠C = 60° and AB = 8, then the length of CD is ______.	4
	<image> As shown in the figure, ∠MCN=45°, and AB∥CD, AC∥BD, E is the intersection point of BE and CN, find the degree measure of ∠DBE.	45
	<image>(2015 Spring • Mid-term Exam of Gaoyou City) As shown in the figure, in rectangle ABCD, E and F are the midpoints of AD and BC, respectively. Connect AF, BE, CE, and DF, intersecting at points M and N. The quadrilateral EMFN is ____.	Rhombus
	<image> Given that the graph of the quadratic function $$y = ax^{2} + bx + c$$ is roughly as shown in the image, the line $$y = bx + c$$ does not pass through the ______ quadrant.	3
	<image> Given that, as shown in the figure, points D and E are on AB and AC respectively. △ABE ≌ △ACD, AB = 5, AD = 2. Find the length of CE.	3
	<image> As shown in the figure, point $$O$$ is the intersection of the diagonals of the rhombus $$ABCD$$, $$DE \parallel AC$$, $$CE \parallel BD$$, connect $$OE$$. Prove: $$OE = BC$$.	OE=BC
	<image>【Problem】As shown in the figure, the paper △ABC is folded along DE, and point A falls at point A_{1}. It is known that ∠1 + ∠2 = 100°. Then ∠A = _______.	50
	<image>As shown in the figure, lines AB and CD intersect at point E, and EF is the angle bisector of ∠BED. Given that ∠DEF = 70°, what is the measure of ∠AED? ___________	40
	<image>On July 28, 1976, a 7.8 magnitude earthquake struck Tangshan, Hebei, China. Most of the buildings collapsed, and 240,000 people suffered casualties. It was later found that the buildings with the least damage were those with wooden structures and triangular roofs, as shown in the diagram. This is due to the function of ____, which is widely used in mechanical manufacturing and construction engineering.	Triangles possess stability
	<image>(Spring 2016•Beijing Final Exam) Place a set of triangle rulers as shown in the figure. If ∠BAC = 31°45′, what is the measure of ∠EAD? _____ degrees.	31°45′
	<image> As shown in the figure, the diagonals $$AC$$ and $$BD$$ of quadrilateral $$ABCD$$ are perpendicular to each other. $$A_{1}$$, $$B_{1}$$, $$C_{1}$$, and $$D_{1}$$ form the midpoint quadrilateral of quadrilateral $$ABCD$$. If $$AC = 8$$ and $$BD = 10$$, then the area of quadrilateral $$A_{1}B_{1}C_{1}D_{1}$$ is ______.	20
	<image>As shown in the figure, taking point O as the center of homothety, the pentagon ABCDE is enlarged to obtain the pentagon A′B′C′D′E′. Knowing that OA = 10 cm and OA′ = 20 cm, the ratio of the perimeter of pentagon ABCDE to the perimeter of pentagon A′B′C′D′E′ is ___.	1:2
	<image>As shown in the figure, \textit{AC} is the diagonal of the rhombus \textit{ABCD}. Points \textit{E} and \textit{F} are on sides \textit{AB} and \textit{AD} respectively, and \textit{AE} = \textit{AF}. Try to prove that △\textit{ACE} ≌ △\textit{ACF}.	△ACE≌△ACF
	<image>The school intends to plant flowers in a rectangular vacant lot. If 10 rose plants are planted per square meter, how many rose seedlings need to be purchased in total?	3750
	<image>As shown in the figure, the rectangle ABCD is folded along BE. If ∠CBA′=30°, then ∠BEA′=_____.	60°
	<image>In the plane Cartesian coordinate system xOy, let the line y=x+2m be tangent to the circle x^2 + y^2 = n^2, where m, n ∈ N* and 0 < \|m - n\| ≤ 1. If the zero of the function f(x) = mx + 1 - n, denoted as x0, falls within the interval (k, k+1), where k ∈ Z, then k = _.	0
	<image>The height and width of the "+" sign are both 10 centimeters. What is its perimeter in centimeters?	40
	<image>As shown in the figure, it is known that AD is the bisector of ∠EAC and AD∥BC. ∠B=30°. Find the degree measures of the following angles: (1) ∠EAD; (2) ∠DAC; (3) ∠C.	30^\circ
	<image>DCEF is a parallelogram. Square FEGH holds an inscribed circle. What is the perimeter of inscribed circle?	13*π

<image>As shown in the figure, A and D are two points on circle O, and BC is the diameter. If ∠D = 35°, then the degree of ∠COA is ________.

70°

<image>ABC is an isosceles triangle where the lengths of AB and BC are equal. CBD is in the form of a sector. What is the total area of shape DBE?

32*√(3)

<image>As shown in the , AB is parallel to CD, DE is perpendicular to AC with the foot of the perpendicular being E, and ∠A is 55°. What is the degree measure of ∠D?

35°

<image>As shown in the figure, △ABC is an equilateral triangle. △AP′B can coincide with △APC after rotation. So: (1) Identify the center of rotation; (2) Determine the degree of the rotation angle; (3) Find the degree of ∠PAP′.

60°

<image>ABCD conforms a square. CBE takes on a sector shape. EBF is identified as a sector. FBG is an isosceles triangle in which FB and BG are equal. Could you specify the length of GB in shape FBG?

13

<image> As shown in the figure, the circle is divided into several equal parts and rearranged into an approximate rectangle. The width of the rectangle is 4 cm, and the length is ( ___ ) cm.

12.56

<image>Side CD transforming into an equilateral triangle beyond the rectangle. DEFG is identified as a parallelogram. IHJK is geometrically a square. In shape IHJK, how long is KJ?

3

<image>ABCD takes the form of a parallelogram. DCEF is a square. FEG is a sector. GEH is an isosceles triangle where GE is the same length as EH. Could you specify the length of HE in shape GEH?

9

<image>Side CD morphs into an equilateral triangle. ECF is identified as a sector. FCG is defined as a sector. Square GCHI has an inner inscribed circle. How much is the perimeter of circle?

36*π

<image> Given: As shown in the figure, in the isosceles trapezoid ABCD, AD ∥ BC, AB = DC, and AC and BD intersect at point O. Prove that OB = OC.

OB=OC

<image>As shown in the figure, it is known that line AB intersects with line CD at point O, OE bisects ∠AOC, ray OF is perpendicular to CD at point O, and ∠BOF = 32°. Find the degree measure of ∠COE.

61°

<image>ACDE is a parallelogram. EDFG conforms a square. What is the total area of the shape EDFG?

18

<image>(2014 Autumn • Mid-term in Hailing District) As shown in the figure, △ABO ≌ △CBO, if ∠A = 85°, ∠ABO = 35°, then the measure of ∠BOC is ____°.

60

<image> As shown in the figure, quadrilateral $$ABCD$$ is a rhombus, $$O$$ is the intersection point of its two diagonals. Three lines passing through point $$O$$ divide the rhombus into shaded and unshaded parts. When the lengths of the two diagonals of the rhombus are $$6$$ and $$8$$ respectively, the area of the shaded part is ______.

12

<image> As shown in the figure, given a rectangular paper AB, point E is the midpoint of AB, point G is a point on BC, and ∠BEG > 60°. Now fold the paper along the line EG so that point B falls at point H on the paper. Connect AH, then the number of angles equal to ∠BEG is (___).

3

<image> The Olympic Park observation tower consists of five independent towers of varying heights arranged in a staggered manner. During a comprehensive practical activity class, a group of students decided to measure the height of the tallest tower using a clinometer (assuming the height of the clinometer is negligible). Their method is as follows: as shown in the figure, they first measure the angle of elevation of the top of the tallest tower $A$ from point $B$ to be $45^{\circ}$. They then walk directly 90 meters towards the base of the tallest tower to reach point $C$, where they again measure the angle of elevation of the top of the tallest tower $A$, which is $58^{\circ}$. Please help them calculate the approximate height $AD$ of the tallest tower. (Reference data: $\sin 58^{\circ} \approx 0.85$, $\cos 58^{\circ} \approx 0.53$, $\tan 58^{\circ} \approx 1.60$)

240

<image>As shown in the figure, points E, F, G, and H are the midpoints of the sides AB, BC, CD, and DA of parallelogram ABCD, respectively. Prove that △BEF ≌ △DGH.

△BEF ≌ △DGH

<image>As shown in the figure, point P is inside ∠AOB. Points $ P_1 $ and $ P_2 $ are the symmetric points of point P with respect to OA and OB, respectively. $ P_1 P_2 $ intersects OA at M and intersects OB at N. If $ P_1 P_2 = 5 $ cm, then the perimeter of △PMN is ______.

5cm

<image>ABCD conforms a parallelogram. DCE is an isosceles triangle where DC and CE are of equal length. GFH takes on a sector shape. In sector GFH, what is the length of arc GH?

π/2

<image>The intersection points of the line \textit{y}=\textit{kx}+\textit{b} with the coordinate axes are shown in the figure. When \textit{y}<0, the range of values for \textit{x} is (__ )

x < 2

<image> As shown in the figure, there are two slides of equal length leaning against a wall. It is known that the height AC of the left slide is equal to the horizontal length DF of the right slide. The sum of the angles ∠ABC and ∠DFE formed by these two slides with the ground is ____ degrees.

90

<image>【Problem】In order to increase the distance tourists walk to view the garden, a straight road with a constant width of \textit{a} meters in the parallelogram-shaped garden, as shown in Figure 1, was changed to a curved road with a constant width of \textit{a} meters, as shown in Figure 2. The remaining areas after the road reconstruction (i.e., the shaded areas in the figure) are denoted as \textit{S}_{1} and \textit{S}_{2}, respectively. Then, \textit{S}_{1} ________ \textit{S}_{2} (fill in with “>”, “=”, or “<”).

=

<image>As shown in the , ∠BAC=110°, if MP and NQ are the perpendicular bisectors of AB and AC respectively, then ∠PAQ=____.

40

<image>As shown in the figure, the main peak of a certain mountain has an elevation of approximately $$600$$ meters. A telecom signal transmission tower $$BC$$ is built on the main peak $$AB$$. Xiao Ming measures the angle of elevation to the peak $$B$$ from point $$P$$ at the foot of the mountain to be $$α$$, and the angle of elevation to the top of the transmission tower $$C$$ to be $$β$$. Given that $$\tan \alpha =\dfrac{3}{5}$$ and $$\tan \beta =\dfrac{5}{8}$$, find the height of the transmission tower $$BC$$.

25

<image>As shown in the figure, AC and BD intersect at O, and AB = DC, AC = BD. Prove that OB = OC.

OB=OC

<image> As shown in the figure, the right-angle vertices of a pair of set squares coincide. (1) Compare the sizes of ∠EOM and ∠FON, and explain the reason; (2) If ∠FOM = 60°, find the degree measure of ∠EON.

120°

<image>ABC is an isosceles triangle with AB having the same length as BC. Side DE developing into an equilateral triangle outside. HGI takes the form of a sector. In sector HGI, how long is arc HI?

12*π

<image>As shown in the figure, points \textit{A}, \textit{B}, and \textit{C} are on circle \textit{O}, \textit{AO} ∥ \textit{BC}, ∠\textit{AOB} = 50°. Then the measure of ∠\textit{OAC} is _________. ___________________________________________________________________________________

25

<image>As shown in the figure, AB=DE, AC=DF, and points E and C are on line BF, with BE=CF. Prove that △ABC≌△DEF.

△ABC≌△DEF

<image>DCEF takes the form of a parallelogram. Side GH transforming into an equilateral triangle beyond the rectangle. IGJK forms a square. Could you specify the length of JG in shape IGJK?

12

<image>As shown in the figure, in triangle △BAC, AB = AC, AB + BC = 13, and the perpendicular bisector MN of side AB intersects AC at D. Find the perimeter of △BCD. ________ _

13

<image>In circle ⊙O, the chord AB = 2 cm, and ∠ACB = 30°, then the diameter of ⊙O is ____ _______ __ cm.__

4

<image>As shown in the figure, ⊙O is the circumcircle of triangle △ABC, and it is known that ∠ABO=50°. Then, the measure of ∠ACB is (______ )

40

<image>DCE is an isosceles triangle in which DC and CE are equal. Square ECFG includes an inscribed circle. What is the entire perimeter length of inscribed circle?

π

<image>As shown in the figure, given ∠O = 35°, and CD is the perpendicular bisector of OA, then the degree of ∠ACB is ______.

70°

<image>The positions of real numbers $ a $ and $ b $ on the number line are shown in the figure. Then $ |a| $ ____ $ |b| $ (fill in with "\>", "\<" or "\=").

>

<image>As shown in the , it is known that ∠COB = 2∠AOC, OD bisects ∠AOB, and ∠AOC = 40°. Find the measure of ∠BOD.

60

<image>As shown in the , in the quadrilateral $$ABCD$$, $$∠1=∠2$$, $$∠D=72^{\circ}$$, then $$∠BCD=$$ ______ .

108°

<image>As shown in the figure, point D is inside the isosceles right triangle ABC, with BC as the hypotenuse. If triangle ABD is rotated counterclockwise about point A to the position of triangle ACD', then ∠ADD' = ______.

45

<image>As shown in the figure, AB = AC, AD ⊥ BC, and the foot of the perpendicular is D. If ∠BAD = 20°, then ∠BAC = ___ °.

40

<image> As shown in the figure, $$\triangle ABC \cong \triangle DEF$$. Based on the information provided in the figure, write $$x =$$ ______ .

20

<image>As shown in the figure, in an equilateral triangle, $ BC = 6 \text{ cm} $, the ray $ AG \parallel BC $, point $ E $ starts moving from point $ A $ along the ray $ AG $ at a speed of $ 1 \text{ cm/s} $. At the same time, point $ F $ starts moving from point $ B $ along the ray $ BC $ at a speed of $ 2 \text{ cm/s} $. Let the movement time be $ t $ $ (s) $. (1) Connect $ EF $. When $ EF $ passes through the midpoint of side $ AC $, find the value of $ t $. (2) During the movement, can the quadrilateral with vertices $ A $, $ C $, $ E $, and $ F $ be a rhombus? If yes, find the value of $ t $; if not, please explain why.

2

<image>As shown in the , on a straight railway, points A and B are 25 km apart. Villages C and D are located such that DA=10 km, CB=15 km, DA is perpendicular to AB at point A, and CB is perpendicular to AB at point B. Now, a transfer station E needs to be constructed on AB such that the distances from villages C and D to station E are equal. How far should station E be built from point A?

15

<image>ABCD takes the form of a parallelogram. DCE forms a sector. ECF is a sector. Can you tell me the length of FC in shape ECF?

4

<image> As shown in the figure, AD is the median of △ABC, ∠ADC=60°. Fold △ADC along the line AD, making point C fall to C'. If BC' = 5, then BC = ____.

10

<image>As shown in the figure, given that ∠1 = 70º and CD∥BE, what is the degree measure of ∠B? (__)

110

<image>As shown in the figure, there are several types of cards A (square), B (square), and C (rectangle). If you want to form a large rectangle with a length of (a+2b) and a width of (a+b), you will need ______ of C type cards.

3

<image>As shown in the figure, the area of the shaded region is ____ square centimeters.

4

<image>ABCD conforms a square. CBEF is geometrically a square. CFG is an isosceles triangle, having CF and FG of the same length. In the isosceles triangle CFG, what are the measures of angle FCG and angle CGF?

67.5

<image> As shown in the diagram, in △ABC, AB = AC, and CD bisects ∠ACB. Given that ∠A = 36°, then the measure of ∠BDC is ____.

72

<image>ABC is an isosceles triangle where AB = BC. CBDE is a parallelogram. EDF is an isosceles triangle where the length of ED equals the length of DF. Side GH curves into a semi-circle. What is the arc length GH in the shape FDGH?

π

<image>As shown in the diagram, a point A on the circumference of a coin coincides with the origin O of the number line. The coin rolls one complete turn along the positive direction of the number line, and point A coincides exactly with a point A' on the number line. If the radius of the coin is 1 unit length, what number does the point A' on the number line represent?

2\pi

<image> As shown in the figure, the coordinates of $$A$$ and $$B$$ are $$(1,0)$$ and $$(0,2)$$, respectively. If the line segment $$AB$$ is translated to $$A_{1}B_{1}$$, then the value of $$a-b$$ is ______ .

0

<image>As shown in the figure, in the parallelogram ABCD, point E is on side BC, and CE:BC=2:3. AC intersects DE at point F. If the area of △AFD is 9, then the area of △EFC is _______ .__

4

<image>If the length of the AD side is 9, the length of the BE side is 22 and the degree of the BEA angle is 55, compute the perimeter of the ABCD parallelogram. Round computations to 2 decimal places.

54.08

<image> (Spring 2011, Guangzhou school-level midterm) As shown in the figure, if ∠1 = 25°, then ∠2 = ____, ∠3 = ____, ∠4 = ____.

155°

<image>In Figure ①, there is a rectangular piece of paper with a length of $2a$ and a width of $2b$. The area of the rectangle is clearly $4ab$. Now, this rectangular piece of paper is cut along the dotted line in the figure, dividing it into 4 smaller rectangles, and then reassembled into a square as shown in Figure ②. (1) In Figure ②, the side length of the shaded square $EFGH$ is: ________________; (2) Observing Figure ②, which shape's area is represented by the algebraic expression $(a - b)^{2}$? How about the algebraic expression $(a + b)^{2}$? (3) Express the area of the shaded square $EFGH$ in Figure ② using two different methods, and write out the equal relationship between the algebraic expressions $(a + b)^{2}$, $(a - b)^{2}$, and $4ab$; (4) Based on the equal relationship in question (3), solve the following problem: If $a + b = 7$ and $ab = 5$, find the value of $(a - b)^{2}$.

a-b

<image> In the parallelogram $$ABCD$$, fold $$\triangle BCD$$ along $$BD$$ so that point $$C$$ falls at point $$E$$, and $$BE$$ intersects $$AD$$ at point $$O$$. Prove that $$OA=OE$$.

OA=OE

<image>As shown in the , OC is a ray inside the angle ∠AOB. Triangle ODE is a triangle with a 60° angle, with its vertex at point O coinciding with the 60°, and side OD is on the ray that exactly bisects ∠AOC, while side OE is on the ray that exactly bisects ∠BOC. Find the degree measure of ∠AOB.

120

<image>As shown in the figure, in the parallelogram $$ABCD$$, $$AB{=}5{,}AD{=}8{,}{DE}$$ bisects $$∠ADC$$. Then $$BE{=}$$______.

3

<image>If the ADEF shape is a combination of a rectangle and a semi-circle and the area of the ADEF shape is 90, compute the perimeter of the ABCD square. Assume $\pi=3.14$. Round computations to 2 decimal places.

50.6

<image>As shown in the figure, there is a square inside a rectangle. Given that AB = 7 cm and CD = 5 cm, find the perimeter of the large rectangle.

24

<image> As shown in the figure, AD and AC are the diameter and chord of circle O respectively, and ∠CAD=30°. OB is perpendicular to AD and intersects AC at point B. If OB=5, then the length of BC is (___ ).

5

<image>DCE is an isosceles triangle with DC and CE being equal in length. ECF is an isosceles triangle, with EC equalling CF. How long is the base EF in the isosceles triangle ECF?

39

<image>DCEF is identified as a parallelogram. FEGH conforms a square. Square HGIJ contains an inscribed circle. What is the area of the entire circle?

1089*π

<image>ABCD is geometrically a square. ADEF is geometrically a square. EDG is in the form of a sector. In shape EDG, what is the length of GD?

24

<image>If the arc length of the blue sector is 17.99, the pink shape is a rectangle where a semi-circle has been removed from one side of it, the perimeter of the pink shape is 44 and the area of the brown square is 64, compute the degree of the angle marked with question mark. Assume $\pi=3.14$. Round computations to 2 decimal places.

86.4

<image>(2010 Autumn • Yanping District Final Exam) As shown in the figure, in the parallelogram ABCD, AC and BD intersect at point O. Given that AB = 8, BC = 6, and the perimeter of △AOB is 18, what is the perimeter of △AOD? _____.

16

<image>As shown in the figure, line AB and line CD intersect at point O. Ray OM bisects ∠AOC, and ON is perpendicular to OM. If ∠AOM = 35°, then the degree measure of ∠CON is (___)

55

<image>As shown in the figure, a right triangle board ABC with a 30° angle has its right-angled vertex A on the line DE, and BC∥DE. Then ∠CAE is equal to (__)

30

<image>As shown in the figure, to measure the height of the flagpole $ AB $, a rope hanging down from the top of the flagpole can be used. When the rope is vertical to the ground, it is found to be $ 1m $ longer than the height of the flagpole. When the rope is stretched to point $ C $ on the ground, the length of $ CB $ is measured to be $ 5m $. Determine the height of the flagpole $ AB $.

12

<image>If the area of the ABCD parallelogram is 54, the ABEF shape is a rectangle where a semi-circle has been removed from one side of it, the length of the BE side is 9, the area of the ABEF shape is 96, the ADHIJ shape is a rectangle where an equilateral triangle has been removed from one side of it, the length of the DH side is 6 and the perimeter of the ADHIJ shape is 42, compute the degree of the BAD angle. Assume $\pi=3.14$. Round computations to 2 decimal places.

18.66

<image>ABC takes the form of a sector. EDFG conforms a square. GFH is an isosceles triangle in which GF and FH are equal. In the isosceles triangle GFH, what are the measures of angle FGH and angle GHF?

45

<image>(2010 Autumn • Shangrao County School Level Midterm) As shown in the figure, △ABC ≌ △ADE, ∠1 = 30°, then ∠BAD = ____.

30°

<image>As shown in the figure, ∠A is an inscribed angle of circle O, and ∠A = 60°. Then the measure of ∠OBC is ____ degrees.

30

<image>ABCD conforms a square. ADE is an isosceles triangle, having AD and DE of the same length. FGH is an equilateral triangle. In the shape HFI, what is the measurement of HI?

√(2)

<image>Side CD transforming into an equilateral triangle beyond the rectangle. DEFG is geometrically a square. FEHI is geometrically a square. What is the measurement of the perimeter of FEHI?

20

<image> As shown in the figure, the area of triangle ABE is 48 square decimeters, and BC = CD = DE. Then the area of the shaded part is ____ square decimeters.

16

<image>Please complete the following proof process. Given: As shown in the figure, $$DE/\!/BC$$, $$BE$$ bisects $$∠ABC$$. Prove: $$∠1=∠3$$. ___ Proof: Because $$BE$$ bisects $$∠ABC$$ (Given), ___ therefore $$∠1=$$_______$$($$_ ________ ______ $$). ___ Also, because $$DE/\!/BC$$ (Given), ___ therefore $$∠2=$$_______$$($$_ ________________ $$). ___ Therefore, $$∠1=∠3$$ ($$ _________________ $$).

∠1=∠3

<image>As shown in the figure, in △ABC, AB = AC, and the altitude AD is drawn on the base BC. Prove: ∠B = ∠C

∠B=∠C

<image>As shown in the figure, in $$\triangle ABC$$, $$BC \perp AC$$, $$CD$$ is the height on side $$AB$$, if $$AB=10cm$$, $$BC=6cm$$, $$AC=8cm$$, then $$CD=$$________

4.8

<image> As shown in the figure, in triangle △ABC, AD is the angle bisector. If ∠B = ∠C = 60° and AB = 8, then the length of CD is ______.

4

<image> As shown in the figure, ∠MCN=45°, and AB∥CD, AC∥BD, E is the intersection point of BE and CN, find the degree measure of ∠DBE.

45

<image>(2015 Spring • Mid-term Exam of Gaoyou City) As shown in the figure, in rectangle ABCD, E and F are the midpoints of AD and BC, respectively. Connect AF, BE, CE, and DF, intersecting at points M and N. The quadrilateral EMFN is ____.

Rhombus

<image> Given that the graph of the quadratic function $$y = ax^{2} + bx + c$$ is roughly as shown in the image, the line $$y = bx + c$$ does not pass through the ______ quadrant.

3

<image> Given that, as shown in the figure, points D and E are on AB and AC respectively. △ABE ≌ △ACD, AB = 5, AD = 2. Find the length of CE.

3

<image> As shown in the figure, point $$O$$ is the intersection of the diagonals of the rhombus $$ABCD$$, $$DE \parallel AC$$, $$CE \parallel BD$$, connect $$OE$$. Prove: $$OE = BC$$.

OE=BC

<image>【Problem】As shown in the figure, the paper △ABC is folded along DE, and point A falls at point A_{1}. It is known that ∠1 + ∠2 = 100°. Then ∠A = _______.

50

<image>As shown in the figure, lines AB and CD intersect at point E, and EF is the angle bisector of ∠BED. Given that ∠DEF = 70°, what is the measure of ∠AED? ___________

40

<image>On July 28, 1976, a 7.8 magnitude earthquake struck Tangshan, Hebei, China. Most of the buildings collapsed, and 240,000 people suffered casualties. It was later found that the buildings with the least damage were those with wooden structures and triangular roofs, as shown in the diagram. This is due to the function of ____, which is widely used in mechanical manufacturing and construction engineering.

Triangles possess stability

<image>(Spring 2016•Beijing Final Exam) Place a set of triangle rulers as shown in the figure. If ∠BAC = 31°45′, what is the measure of ∠EAD? _____ degrees.

31°45′

<image> As shown in the figure, the diagonals $$AC$$ and $$BD$$ of quadrilateral $$ABCD$$ are perpendicular to each other. $$A_{1}$$, $$B_{1}$$, $$C_{1}$$, and $$D_{1}$$ form the midpoint quadrilateral of quadrilateral $$ABCD$$. If $$AC = 8$$ and $$BD = 10$$, then the area of quadrilateral $$A_{1}B_{1}C_{1}D_{1}$$ is ______.

20

<image>As shown in the figure, taking point O as the center of homothety, the pentagon ABCDE is enlarged to obtain the pentagon A′B′C′D′E′. Knowing that OA = 10 cm and OA′ = 20 cm, the ratio of the perimeter of pentagon ABCDE to the perimeter of pentagon A′B′C′D′E′ is ___.

1:2

<image>As shown in the figure, \textit{AC} is the diagonal of the rhombus \textit{ABCD}. Points \textit{E} and \textit{F} are on sides \textit{AB} and \textit{AD} respectively, and \textit{AE} = \textit{AF}. Try to prove that △\textit{ACE} ≌ △\textit{ACF}.

△ACE≌△ACF

<image>The school intends to plant flowers in a rectangular vacant lot. If 10 rose plants are planted per square meter, how many rose seedlings need to be purchased in total?

3750

<image>As shown in the figure, the rectangle ABCD is folded along BE. If ∠CBA′=30°, then ∠BEA′=_____.

60°

<image>In the plane Cartesian coordinate system xOy, let the line y=x+2m be tangent to the circle x^2 + y^2 = n^2, where m, n ∈ N* and 0 < |m - n| ≤ 1. If the zero of the function f(x) = mx + 1 - n, denoted as x0, falls within the interval (k, k+1), where k ∈ Z, then k = _.

0

<image>The height and width of the "+" sign are both 10 centimeters. What is its perimeter in centimeters?

40

<image>As shown in the figure, it is known that AD is the bisector of ∠EAC and AD∥BC. ∠B=30°. Find the degree measures of the following angles: (1) ∠EAD; (2) ∠DAC; (3) ∠C.

30^\circ

<image>DCEF is a parallelogram. Square FEGH holds an inscribed circle. What is the perimeter of inscribed circle?

13*π