Sicong/RL_toy_math_2 · Datasets at Hugging Face

images images listlengths 1 1	problem stringlengths 7 3.71k	answer stringlengths 1 37
	<image>ACD takes the form of a sector. DCE forms a sector. Side FG takes the shape of an equilateral triangle. In the shape ECFHG, what is the measurement of GH?	110
	<image>(2013 Spring•Zhongshan Final Exam) As shown in the , if ∠C = 70°, ∠A = 30°, and ∠D = 110°, then ∠1 + ∠2 equals ____ degrees.	210
	<image> Person A rides a bicycle from point $$A$$ to point $$B$$, while Person B walks from point $$B$$ to point $$A$$, as shown in the figure. $$y_{A}$$ and $$y_{B}$$ represent the relationships between the distances $$y(km)$$ of Person A and Person B from point $$A$$ and the elapsed time $$x(h)$$, respectively. The lines $$y_{A}$$ and $$y_{B}$$ intersect at point $$M$$. $$(1)$$ Find the function relationship between $$y_{A}$$ and $$x$$ (there is no need to specify the range of the variable $$x$$); $$(2)$$ Find the distance between points $$A$$ and $$B$$.	10
	<image>ABCD conforms a parallelogram. DCE is in the form of a sector. Side FG expands into a semi-circle. What is the measurement of FG in the shape ECFG?	44*π
	<image>ABC is identified as a sector. Arc FG is in the shape of a semi-circle. What is the total surface area of the shape EDFG?	216 - 81*π/2
	<image>ABC takes on a sector shape. CBD takes on a sector shape. DBEF takes the form of a parallelogram. Side GH shapes into an equilateral triangle. In shape FEGIH, what is the length of HI?	2
	<image>ABCD is geometrically a parallelogram. DCEF conforms a square. DFG is an isosceles triangle where the sides DF and FG are identical in length. Side HI continues into a semi-circle within the rectangle. What is the length of HI in shape GFHI?	81*π/2
	<image>As shown in the image, four identical rectangles and a small square are used to form a large square with a side length of 16 decimetres. What is the perimeter of each rectangle?	32
	<image> (Autumn 2010, Dafeng City Final Exam) As shown in the diagram, in right triangle ABC, ∠ACB = 90°, CD is the median on hypotenuse AB, P is the midpoint of CD, if point P lies on the circumference of the circle with AC as its diameter, then ∠A = ____.	60
	<image> A trapezoidal lawn was built in the Kangle Community. For the convenience of residents, a 1-meter-wide path was constructed within the lawn (as shown in the figure). It costs 18 yuan per square meter to plant the lawn. How much did it cost to plant this lawn?	3366
	<image>DBE is an isosceles triangle, having DB and BE of the same length. Could you provide the length of the base DE in the isosceles triangle DBE?	1/3
	<image>As shown in the figure, on a rectangular field ABCD with a length of 15 meters and a width of 10 meters, there are three paths of the same width, one of which is parallel to AD and the other two are parallel to AB. The remaining area is a lawn. Given that the total area of the lawn is 126 square meters, find the width of the paths.	1
	<image>By measurement, the perimeter of the figure below is ____.	5 cm
	<image>The diagram (which is not drawn to scale) shows a rectangle $A B C D$ and a square $P Q R S$, in which $P Q=B C=6 \mathrm{~cm}$ and $C D=10 \mathrm{~cm} . P Q$ is parallel to $A B$. The shaded area is half the area of $A B C D$. <image1> What is the length, in $\mathrm{cm}$, of $P X$ ?	1
	<image>DCE is a sector. Side FG develops into an equilateral triangle inside the rectangle. How can we calculate the perimeter of ECFHG?	429
	<image>As shown in the figure, in triangle △ABC, ∠C=90°, DE is the perpendicular bisector of AB, and ∠CAD:∠CAB=1:5. Find the measure of ∠B in degrees.	40
	<image>ABC is an isosceles triangle with AB having the same length as BC. CBDE conforms a square. EDFG is a square. What is the perimeter length of EDFG?	72
	<image>As shown in the figure, in quadrilateral $$ABCD$$, $$∠A=70^{\circ}$$. Rotate quadrilateral $$ABCD$$ clockwise around vertex $$B$$ to form quadrilateral $$A_{1}BC_{1}D_{1}$$. When $$C_{1}D_{1}$$ first passes through vertex $$C$$, the rotation angle $$∠ABA_{1}=$$_______.	40
	<image>(2015 Spring • Qingyang District Final Exam) As shown in the figure, in △ABC, ∠A = 90°, point D is on side AC, and DE ∥ BC. If ∠ADE = 139°, then the measure of ∠B is ____.	49
	<image>ACDE takes the form of a parallelogram. EDF is identified as a sector. FDG is an isosceles triangle, with FD equalling DG. In isosceles triangle FDG, what is the measure of angle DFG and angle FGD?	45
	<image>As shown in the , AB is parallel to CD, and ∠1 measures 120°. (1) Find the degree measure of ∠A; (2) If ∠C equals ∠A, are AD and BC parallel? Why?	120°
	<image>ABCD conforms a square. Side EF transforms into an equilateral triangle. FGHI is geometrically a parallelogram. Arc JK takes the shape of a semi-circle. What is the length of JK in shape IHJK?	9*π/2
	<image>CBD is defined as a sector. Arc EF is in the shape of a semi-circle. Please provide the length of EF in shape DBEF.	36√(3)π
	<image>According to the diagram, find the measures of ∠2 and ∠3. ∠1=38°, ∠2=____, ∠3=____.	52°
	<image>Known: As shown in the , in triangle ABC, ED is the perpendicular bisector of AB, ∠EBC = 24°, ∠C = 72°, try to find the measure of ∠A.	42
	<image>CBDE forms a square. Side FG continuing into an equilateral triangle outside the rectangle. GHI is an isosceles triangle where the length of GH equals the length of HI. Please provide the measures of angle HGI and angle GIH in the isosceles triangle GHI.	67.5
	<image>As shown in the figure, AB is a chord of circle O. Given that ∠A = 35°, find the measure of ∠AOB in degrees.	110
	<image>ABC is geometrically a sector. CBD is in the form of a sector. FEGH takes the form of a parallelogram. What is the entire perimeter length of FEGH?	2 + 2*√(2)
	<image>ABC is a sector. CBD is an isosceles triangle, with CB equalling BD. Side EF forms the shape of an equilateral triangle. FGH is an isosceles triangle, having FG and GH of the same length. How would you calculate the total area of the shape FGH?	1/2
	<image>As shown in the figure, circles with a radius of 1 are drawn with the vertices of a triangle, quadrilateral, ..., n-sided polygon as centers, and the circles do not intersect each other. The area of the overlap between the triangle and the circles is denoted as \(S_{3}\), the area of the overlap between the quadrilateral and the circles is denoted as \(S_{4}\), ..., and the area of the overlap between the n-sided polygon and the circles is denoted as \(S_{n}\). What is the value of \(S_{90}\) (keep the result in terms of \(\pi\))?	44π
	<image> As shown in the figure, the schematic diagram is a computer motherboard, with each corner being a right angle. The data is as shown in the figure ($$units:$$ mm), then the perimeter of the motherboard is ______ .	96 \text{mm
	<image> A rectangle is divided into four smaller rectangles by two lines. The areas of three of the smaller rectangles are given in the diagram. Find the area of the shaded part (unit: cm^{2}).	37.5
	<image> There are two squares with a side length of 8 centimeters, overlapping at one corner (as shown in the image). Find the area of the non-overlapping part.	96
	<image>ABC is in the form of a sector. CBDE forms a square. Side FG is prolonged into an equilateral triangle within the rectangle. How can we determine the area of the entire shape DBFHG?	121 - 121*√(3)/4
	<image>ABC is an isosceles triangle where the lengths of AB and BC are equal. CBD is an isosceles triangle in which CB equals BD in length. DBEF forms a square. DFG conforms a sector. How long is arc DG in sector DFG?	2*π/3
	<image>Side CD developing into an equilateral triangle outside. ECFG is a square. FCHI is a parallelogram. IHJ is an isosceles triangle, having IH and HJ of the same length. What is the measurement of the total area of IHJ?	3528
	<image>ECF is an isosceles triangle where EC = CF. What is the total area of shape FCGH?	7470*√(3)
	<image>ABC is an isosceles triangle with AB having the same length as BC. CBD is an isosceles triangle where CB is the same length as BD. Side EF extending into a semi-circular shape beyond the rectangle. How do you calculate the area of the shape DBEF?	169*π/8 + 312
	<image>ABCD forms a square. EFG describes an equilateral triangle. How much area does the shape FGH cover?	9*√(3)/2
	<image>As shown in the figure: it is a diagram of a rectangular billiard table surface ABCD, with AB=2m and AD=1.5m. Point E is an arbitrary point on edge AD. A ball starts from point E, collides with three edges, and returns to point E (via E to F to G to H to E). Ignoring the size of the ball, the total length of the path traveled by the ball is ________.	5
	<image>ABC is identified as a sector. CBDE is geometrically a square. EDF takes on a sector shape. FDGH takes the form of a parallelogram. What is the measurement of the total area of FDGH?	3640*√(2)
	<image>ABCD is a parallelogram. Square DEFG includes a circle inscribed within. What is the side length FE in the shape DEFG?	50*√(2)
	<image>As shown in the figure, the right triangle △ABC (∠ABC=90°) is translated to the right along the side CB to obtain △DFE. DE intersects AB at point G. Given: DF = 9 cm, CE = 4 cm, and AG = 4 cm, find the lengths of BF = ______ cm and BG = ______ cm.	4
	<image>If the area of the green parallelogram is 90, compute the degree of the angle marked with question mark. Round computations to 2 decimal places.	47.73
	<image>As shown in the figure, a straight line \( l \) passes through vertex \( B \) of the square \( ABCD \). The distances from points \( A \) and \( C \) to the straight line \( l \) are \( a \) and \( b \) respectively. Then the area of the square is ___.	a^2+b^2
	<image>As shown in the figure, OAB is a sector with a radius of 6, AC is tangent to arc AB at point A and intersects the extension line of OB at point C. If arc AB = 3 cm and AC = 4 cm, find the area of the shaded region.	3
	<image>(2013 Fall • Wenling City School Monthly Exam) As shown in the diagram, in △ABC, CE and BF are two altitudes. If ∠A = 70°, ∠BCE = 25°, then the measure of ∠EBF is ____, and the measure of ∠FBC is ____.	20^\circ\ and\ 45^\circ
	<image>As shown in the figure, AC = AD, ∠BAC = ∠BAD, and point E is on AB. (1) Can you identify a pair of congruent triangles; (2) Please name a pair of congruent triangles and explain the reason.	\text{△ACB \cong △ADB
	<image>As shown in the figure, in triangle $$\triangle ABC$$, $$AB=AC$$, $$BC=BD$$, and $$AD=DE=EB$$, what is the measure of angle $$\angle A$$ in degrees?	45
	<image>(2016 Spring • Changle City Midterm) As shown in the figure, in the equilateral triangle ABC, point D and E are the midpoints of AB and AC respectively. Then the measure of ∠DEC is (__) degrees.	120°
	<image>ABC takes the form of a sector. Side DE extends into an equilateral triangle outside the rectangle. Side GH is forming an equilateral triangle. What is the length of IG in shape FDGIH?	11
	<image>In the , two monkeys are at point B which is 10 meters high on a tree. To compete for the fruit at point A by the pond, one monkey climbs down the tree and runs to point A, which is 20 meters away from point C. The other monkey climbs to the top of the tree at point D and then jumps directly to point A, the distance being measured in a straight line. If both monkeys travel the same distance, find the height of the tree.	15
	<image>If you use a 145 cm long wooden strip to make a frame for your certificate (with a total loss of 4 cm at the joints). (1) Is this strip long enough? (2) To cover the certificate with a piece of glass, at least how large should the glass be?	Enough
	<image>As shown in the , in triangle ABC, ∠ACB = 90°, ∠A = 20°. Triangle ABC is rotated around point C in the counterclockwise direction by an angle α to the position of triangle A′B′C′, where A′ and B′ are the corresponding points of A and B, respectively, B is on A′B′, and CA′ intersects AB at D. Then, the degree measure of ∠BDC is ____ degrees.	60
	<image>ABCD is a parallelogram. FEG takes the form of a sector. GEH is an isosceles triangle where GE = EH. I need to know the length of the base GH in the isosceles triangle GEH.	22*√(2)
	<image>As shown in the figure, in the square \(ABCD\), \(AB = 3\), point \(E\) is on the side \(CD\) such that \(CD = 3DE\). Fold \(\triangle ADE\) along \(AE\) to \(\triangle AFE\). Extend \(EF\) to intersect \(BC\) at point \(G\). Connect \(AG\) and \(CF\). The following conclusions are given: \(①\) Point \(G\) is the midpoint of \(BC\); \(②\) \(FG = FC\); \(③\) \(\angle GAE = 45^{\circ}\); \(④\) \(\angle FCG = 60^{\circ}\). Among these, the correct statements are ____________.	①③
	<image>DCE is an isosceles triangle where the length of DC equals the length of CE. Square ECFG is inscribed with a circle. In the shape ECFG, what is the measurement of EG?	22
	<image>In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is $X$, in centimeters?	5
	<image> As shown in the figure, the slope of _$$AB$$ is _$$1$$:$$3.$$ If the slope length _$$AB$$ is 10m, then the height of the slope _$$BC$$ is ______$$m$$.	\sqrt{10
	<image>【Problem】As shown in the figure, AD⊥BC, CE⊥BC, CH⊥AB, BG⊥AC. In △ABC, the altitude from the side BC is (_____) Choices: (A) Line segment CE (B) Line segment CH (C) Line segment AD (D) Line segment BG	Line segment AD
	<image>As shown in the , line AB intersects with line CD at point O. EO is perpendicular to CD, and ∠BOE is 48°. The degree measure of ∠AOD is (___).	138
	<image>ABC takes on a sector shape. GHI shapes into an equilateral triangle. What is the overall perimeter of FEGIH?	22*√(2) + 33
	<image>As shown in the , in triangle $$\triangle ABC$$, $$∠1=100^{\circ}$$, $$∠C=80^{\circ}$$, $$∠2= \dfrac{1}{2}∠3$$, and $$BE$$ bisects $$∠ABC$$. Find the measure of $$∠4$$.	45^{\circ
	<image>As shown in the figure, the radius of circle $$⊙O$$ is $$10$$. The length of chord $$AB$$ is $$12$$. The line segment $$OD$$ is perpendicular to $$AB$$, intersecting $$AB$$ at point $$D$$ and intersecting circle $$⊙O$$ at point $$C$$. Then, $$OD =$$ ______ , $$CD =$$ ______ .	8
	<image>(Spring 2015, Final Exam in Shengzhou City) As shown in the figure, in the parallelogram ▱ABCD, AB = 8 cm, AD = 5 cm, the bisector of ∠BAD intersects CD at point E, and the bisector of ∠ABC intersects CD at point F. Then, the length of the segment EF is ____ cm.	2
	<image>As shown in the figure, the distance between two adjacent points in the horizontal and vertical directions is m. If the area of quadrilateral ABCD is 23, then the area of pentagon EFGHI is ____.	28
	<image> As shown in the figure, a rectangular piece of paper ABCD is folded along EF, and points C and D fall on positions C′ and D′ respectively. EC′ intersects AD at point G. Given that ∠EFG=58°, then ∠BEG=(___).	64
	<image>Uncle Wang wants to pave the train station square with colored tiles (as shown in the image), leaving a square flower bed with a perimeter of 28 meters. How many square meters of colored tiles are needed?	4001
	<image>ABC forms a sector. CBD is an isosceles triangle where the sides CB and BD are identical in length. DBE conforms a sector. In the sector DBE, what is the measurement of arc DE?	99*π/2
	<image>(2015 Spring•Tongliao Final Exam) As shown in the figure, if AB∥DC, ∠1=39°, and ∠C and ∠D are complementary, then ∠D=____, ∠B=____.	39°, 129°
	<image> A park is planning to build a rectangular lawn, with a length of 30 meters and a width of 20 meters. A cross-path as shown in the figure will be constructed on the lawn, with the width of the path being x meters. Answer the following questions: (1) What is the area of the cross-path in square meters? (2) If the width of the path is 2 meters, what is the area of the lawn (the shaded part) in square meters?	50x - x^2
	<image>As shown in the figure, in $$\triangle ABC$$, $$∠ABC = 50º$$, $$∠ACB = 75º$$, point $$O$$ is the incenter of $$\triangle ABC$$, then the measure of $$∠BOC$$ is ________ .	117.5^\circ
	<image>If the length of the AD side is 17, the area of the ABCD parallelogram is 114, the length of the BE side is 7, the degree of the FEB angle is 70, the adjacent angles BEA and FEB are complementary, the length of the AG side is 14 and the diagonal of the AGHE rectangle is 19, compute the degree of the DAB angle. Round computations to 2 decimal places.	90
	<image>ABC forms a sector. CBD is an isosceles triangle, having CB and BD of the same length. DBEF conforms a square. Side GH forms an equilateral triangle inside the rectangle. How do you find the perimeter of EBGIH?	352
	<image>ABCD conforms a square. Side GH transitioning into an equilateral triangle outside. Can you calculate the area of the entire shape FEGIH?	400*√(3) + 3000
	<image>As shown in , to bend the angle iron (Figure ①) into a 145° steel frame (Figure ②), the notch cut out on the angle iron (dotted line in Figure ①) should be (__ degrees).	35
	<image>ABC takes the form of a sector. CBDE is geometrically a parallelogram. EDF takes the form of a sector. Side GH converting into a semi-circle outside. Can you calculate the area of the entire shape FDGH?	8281*π/8 + 4641
	<image>As shown in the figure, all squares have their centers at the origin, and all sides are parallel to either the $$x$$-axis or the $$y$$-axis. From the inside out, their side lengths are $$2$$, $$4$$, $$6$$, $$8$$, $$...$$, and their vertices are denoted sequentially as $$A_{1}$$, $$A_{2}$$, $$A_{3}$$, $$A_{4}$$, $$...$$. What are the coordinates of vertex $$A_{2017}$$?	(-505,-505)
	<image>DCE is in the form of a sector. ECFG conforms a parallelogram. Side HI arcs into a semi-circle inside the rectangle. In shape GFHI, what is the length of HI?	48*π
	<image>Please find the areas of the following three figures in the grid below. Each small square represents 1 square centimeter, and any incomplete square should be counted as half a square. (1) What is the approximate area of the butterfly? (2) What is the approximate area of the dragonfly? (3) What is the approximate area of the maple leaf?	20
	<image>ABCD conforms a square. FEGH is geometrically a square. How much is the area of the entire shape FEGH?	81
	<image>In △ABC, AB = AC, and DE is the perpendicular bisector of AB, intersecting AB and AC at D and E respectively. If ∠A = 40°, find the measure of ∠EBC	30
	<image>As shown in the figure, through the midpoint E of the edge AD of the tetrahedron ABCD, a section EFGH is made parallel to edges AB and CD. If AB = 4 and CD = 6, then the perimeter of the section EFGH is _______________.	10
	<image>Find the area of each of the following shapes.	25cm^2
	<image>As shown in the figure, in △ABC, BD is the angle bisector of ∠ABC, DE ∥ BC, intersecting AB at E. Given that ∠A = 60° and ∠BDC = 95°, what is the measure of ∠BED? (_ )	110
	<image>As shown in the figure, line segment $$OC$$ is a ray inside angle $$∠AOB$$, and $$OM$$ and $$ON$$ bisect angles $$∠AOC$$ and $$∠BOC$$, respectively. (1) If $$∠AOC={120}^{0}$$ and $$∠BOC={30}^{0}$$, find the measure of $$∠MON$$; (2) If $$∠AOC=α$$ and $$∠BOC=β$$, express $$∠MON$$ using algebraic expressions involving $$α$$ and $$β$$, and directly state the quantitative relationship between $$∠AOB$$ and $$∠MON$$.	75^\circ
	<image>Cut a square piece of colored paper with a side length of 6 cm into a "Z" shape, as shown in the image. Find the area of the "Z" shape.	14.84
	<image> As shown in the figure, the exterior angle of △ADC is ____.	∠ADB, ∠EAC
	<image>ACD is a sector. DCE is an isosceles triangle with DC and CE being equal in length. What is the perimeter length of DCE?	48 + 48*√(2)
	<image>ABCD is a parallelogram. DCEF conforms a square. FEGH conforms a square. What is the side length GE in the shape FEGH?	52
	<image>As shown in the figure, a student uses a right triangle with a 60° angle to estimate the height of the school flagpole AB. He places the horizontal leg of the 60° angle on a 1.5-meter high stand CD, aligning the hypotenuse of the triangle with the top of the flagpole. He measures the distance from D to B as 5 meters. The height of the flagpole AB is approximately ______ meters (accurate to 1 meter, \(\frac{} = 1.73\)).	10
	<image>ABCD is identified as a parallelogram. FEG is in the form of a sector. GEHI is identified as a parallelogram. How much area does the shape GEHI cover?	√(3)/2
	<image>(2010 Tianhe District Mock Exam) As shown in the figure, a square ABCD with a side length of 2 cm is cut along its diagonal AC, and then △ABC is translated along BC to obtain △A′B′C′. If the overlapping area of the two triangles is denoted as S, the maximum value of S is ___ cm^{2}.	1
	<image>If the brown shape is a rectangle where an equilateral triangle has been removed from one side of it, the area of the brown shape is 114, the area of the lime sector is 56.52 and the angle $\theta$ and the adjacent 30 degree angle are complementary, compute the length of the side of the brown shape marked with question mark. Assume $\pi=3.14$. Round computations to 2 decimal places.	15.48
	<image>(2014 Autumn • Ouhai District Midterm Exam) As shown in Figure 1, this is a diagrammatic representation of the famous ancient Chinese "Zhao Shuang's Inscribed Square Diagram". It is formed by four congruent right-angled triangles. If the shorter leg BC = 2.5, and the longer legs of the four right triangles are each extended outward by an equal amount, resulting in the "Mathematical Windmill" shown in Figure 2, and if the perimeter of △BCD is 15, then the outer perimeter of this windmill is ____.	38
	<image>As shown in the figure, $$∠1 = ∠3$$, and $$∠1 + ∠2 = 180^{\circ}$$. $$(1)$$ Determine the positional relationship between $$GF$$ and $$CB$$, and explain the reason;___ $$(2)$$ If $$BF ⊥ AC$$ and $$∠2 = 150^{\circ}$$, find the degree measure of $$∠AFG$$.$$__	60°
	<image>A math extracurricular group measured the height of the sculpture CD at Jinhu Plaza. They measured the angle of elevation to point D at the top of the sculpture from point A on the ground to be 30°. Moving 18 meters forward towards the base of the sculpture to point B, they measured the angle of elevation to be 45° (as shown in the image). Find the height of the sculpture CD. (accurate to 0.01 meters)	24.59
	<image>EFG forms a triangle with all sides equal. GEH takes on a sector shape. Square HEIJ is inscribed with a circle. What is the total boundary length of circle?	π
	<image>As shown in the figure, points A, B, and C are on the circumference of circle O. If ∠ABC = 40°, then the degree measure of ∠AOC is _________. __	80
	<image>DEF creates an equilateral triangle. EFGH forms a square. What would the area of the entire shape EHI be?	3/2

<image>ACD takes the form of a sector. DCE forms a sector. Side FG takes the shape of an equilateral triangle. In the shape ECFHG, what is the measurement of GH?

110

<image>(2013 Spring•Zhongshan Final Exam) As shown in the , if ∠C = 70°, ∠A = 30°, and ∠D = 110°, then ∠1 + ∠2 equals ____ degrees.

210

<image> Person A rides a bicycle from point $$A$$ to point $$B$$, while Person B walks from point $$B$$ to point $$A$$, as shown in the figure. $$y_{A}$$ and $$y_{B}$$ represent the relationships between the distances $$y(km)$$ of Person A and Person B from point $$A$$ and the elapsed time $$x(h)$$, respectively. The lines $$y_{A}$$ and $$y_{B}$$ intersect at point $$M$$. $$(1)$$ Find the function relationship between $$y_{A}$$ and $$x$$ (there is no need to specify the range of the variable $$x$$); $$(2)$$ Find the distance between points $$A$$ and $$B$$.

10

<image>ABCD conforms a parallelogram. DCE is in the form of a sector. Side FG expands into a semi-circle. What is the measurement of FG in the shape ECFG?

44*π

<image>ABC is identified as a sector. Arc FG is in the shape of a semi-circle. What is the total surface area of the shape EDFG?

216 - 81*π/2

<image>ABC takes on a sector shape. CBD takes on a sector shape. DBEF takes the form of a parallelogram. Side GH shapes into an equilateral triangle. In shape FEGIH, what is the length of HI?

2

<image>ABCD is geometrically a parallelogram. DCEF conforms a square. DFG is an isosceles triangle where the sides DF and FG are identical in length. Side HI continues into a semi-circle within the rectangle. What is the length of HI in shape GFHI?

81*π/2

<image>As shown in the image, four identical rectangles and a small square are used to form a large square with a side length of 16 decimetres. What is the perimeter of each rectangle?

32

<image> (Autumn 2010, Dafeng City Final Exam) As shown in the diagram, in right triangle ABC, ∠ACB = 90°, CD is the median on hypotenuse AB, P is the midpoint of CD, if point P lies on the circumference of the circle with AC as its diameter, then ∠A = ____.

60

<image> A trapezoidal lawn was built in the Kangle Community. For the convenience of residents, a 1-meter-wide path was constructed within the lawn (as shown in the figure). It costs 18 yuan per square meter to plant the lawn. How much did it cost to plant this lawn?

3366

<image>DBE is an isosceles triangle, having DB and BE of the same length. Could you provide the length of the base DE in the isosceles triangle DBE?

1/3

<image>As shown in the figure, on a rectangular field ABCD with a length of 15 meters and a width of 10 meters, there are three paths of the same width, one of which is parallel to AD and the other two are parallel to AB. The remaining area is a lawn. Given that the total area of the lawn is 126 square meters, find the width of the paths.

1

<image>By measurement, the perimeter of the figure below is ____.

5 cm

<image>The diagram (which is not drawn to scale) shows a rectangle $A B C D$ and a square $P Q R S$, in which $P Q=B C=6 \mathrm{~cm}$ and $C D=10 \mathrm{~cm} . P Q$ is parallel to $A B$. The shaded area is half the area of $A B C D$. <image1> What is the length, in $\mathrm{cm}$, of $P X$ ?

1

<image>DCE is a sector. Side FG develops into an equilateral triangle inside the rectangle. How can we calculate the perimeter of ECFHG?

429

<image>As shown in the figure, in triangle △ABC, ∠C=90°, DE is the perpendicular bisector of AB, and ∠CAD:∠CAB=1:5. Find the measure of ∠B in degrees.

40

<image>ABC is an isosceles triangle with AB having the same length as BC. CBDE conforms a square. EDFG is a square. What is the perimeter length of EDFG?

72

<image>As shown in the figure, in quadrilateral $$ABCD$$, $$∠A=70^{\circ}$$. Rotate quadrilateral $$ABCD$$ clockwise around vertex $$B$$ to form quadrilateral $$A_{1}BC_{1}D_{1}$$. When $$C_{1}D_{1}$$ first passes through vertex $$C$$, the rotation angle $$∠ABA_{1}=$$_______.

40

<image>(2015 Spring • Qingyang District Final Exam) As shown in the figure, in △ABC, ∠A = 90°, point D is on side AC, and DE ∥ BC. If ∠ADE = 139°, then the measure of ∠B is ____.

49

<image>ACDE takes the form of a parallelogram. EDF is identified as a sector. FDG is an isosceles triangle, with FD equalling DG. In isosceles triangle FDG, what is the measure of angle DFG and angle FGD?

45

<image>As shown in the , AB is parallel to CD, and ∠1 measures 120°. (1) Find the degree measure of ∠A; (2) If ∠C equals ∠A, are AD and BC parallel? Why?

120°

<image>ABCD conforms a square. Side EF transforms into an equilateral triangle. FGHI is geometrically a parallelogram. Arc JK takes the shape of a semi-circle. What is the length of JK in shape IHJK?

9*π/2

<image>CBD is defined as a sector. Arc EF is in the shape of a semi-circle. Please provide the length of EF in shape DBEF.

36*√(3)*π

<image>According to the diagram, find the measures of ∠2 and ∠3. ∠1=38°, ∠2=____, ∠3=____.

52°

<image>Known: As shown in the , in triangle ABC, ED is the perpendicular bisector of AB, ∠EBC = 24°, ∠C = 72°, try to find the measure of ∠A.

42

<image>CBDE forms a square. Side FG continuing into an equilateral triangle outside the rectangle. GHI is an isosceles triangle where the length of GH equals the length of HI. Please provide the measures of angle HGI and angle GIH in the isosceles triangle GHI.

67.5

<image>As shown in the figure, AB is a chord of circle O. Given that ∠A = 35°, find the measure of ∠AOB in degrees.

110

<image>ABC is geometrically a sector. CBD is in the form of a sector. FEGH takes the form of a parallelogram. What is the entire perimeter length of FEGH?

2 + 2*√(2)

<image>ABC is a sector. CBD is an isosceles triangle, with CB equalling BD. Side EF forms the shape of an equilateral triangle. FGH is an isosceles triangle, having FG and GH of the same length. How would you calculate the total area of the shape FGH?

1/2

<image>As shown in the figure, circles with a radius of 1 are drawn with the vertices of a triangle, quadrilateral, ..., n-sided polygon as centers, and the circles do not intersect each other. The area of the overlap between the triangle and the circles is denoted as $S_{3}$, the area of the overlap between the quadrilateral and the circles is denoted as $S_{4}$, ..., and the area of the overlap between the n-sided polygon and the circles is denoted as $S_{n}$. What is the value of $S_{90}$ (keep the result in terms of $\pi$)?

44π

<image> As shown in the figure, the schematic diagram is a computer motherboard, with each corner being a right angle. The data is as shown in the figure ($$units:$$ mm), then the perimeter of the motherboard is ______ .

96 \text{mm

<image> A rectangle is divided into four smaller rectangles by two lines. The areas of three of the smaller rectangles are given in the diagram. Find the area of the shaded part (unit: cm^{2}).

37.5

<image> There are two squares with a side length of 8 centimeters, overlapping at one corner (as shown in the image). Find the area of the non-overlapping part.

96

<image>ABC is in the form of a sector. CBDE forms a square. Side FG is prolonged into an equilateral triangle within the rectangle. How can we determine the area of the entire shape DBFHG?

121 - 121*√(3)/4

<image>ABC is an isosceles triangle where the lengths of AB and BC are equal. CBD is an isosceles triangle in which CB equals BD in length. DBEF forms a square. DFG conforms a sector. How long is arc DG in sector DFG?

2*π/3

<image>Side CD developing into an equilateral triangle outside. ECFG is a square. FCHI is a parallelogram. IHJ is an isosceles triangle, having IH and HJ of the same length. What is the measurement of the total area of IHJ?

3528

<image>ECF is an isosceles triangle where EC = CF. What is the total area of shape FCGH?

7470*√(3)

<image>ABC is an isosceles triangle with AB having the same length as BC. CBD is an isosceles triangle where CB is the same length as BD. Side EF extending into a semi-circular shape beyond the rectangle. How do you calculate the area of the shape DBEF?

169*π/8 + 312

<image>ABCD forms a square. EFG describes an equilateral triangle. How much area does the shape FGH cover?

9*√(3)/2

<image>As shown in the figure: it is a diagram of a rectangular billiard table surface ABCD, with AB=2m and AD=1.5m. Point E is an arbitrary point on edge AD. A ball starts from point E, collides with three edges, and returns to point E (via E to F to G to H to E). Ignoring the size of the ball, the total length of the path traveled by the ball is ________.

5

<image>ABC is identified as a sector. CBDE is geometrically a square. EDF takes on a sector shape. FDGH takes the form of a parallelogram. What is the measurement of the total area of FDGH?

3640*√(2)

<image>ABCD is a parallelogram. Square DEFG includes a circle inscribed within. What is the side length FE in the shape DEFG?

50*√(2)

<image>As shown in the figure, the right triangle △ABC (∠ABC=90°) is translated to the right along the side CB to obtain △DFE. DE intersects AB at point G. Given: DF = 9 cm, CE = 4 cm, and AG = 4 cm, find the lengths of BF = ______ cm and BG = ______ cm.

4

<image>If the area of the green parallelogram is 90, compute the degree of the angle marked with question mark. Round computations to 2 decimal places.

47.73

<image>As shown in the figure, a straight line $ l $ passes through vertex $ B $ of the square $ ABCD $. The distances from points $ A $ and $ C $ to the straight line $ l $ are $ a $ and $ b $ respectively. Then the area of the square is ___.

a^2+b^2

<image>As shown in the figure, OAB is a sector with a radius of 6, AC is tangent to arc AB at point A and intersects the extension line of OB at point C. If arc AB = 3 cm and AC = 4 cm, find the area of the shaded region.

3

<image>(2013 Fall • Wenling City School Monthly Exam) As shown in the diagram, in △ABC, CE and BF are two altitudes. If ∠A = 70°, ∠BCE = 25°, then the measure of ∠EBF is ____, and the measure of ∠FBC is ____.

20^\circ\ and\ 45^\circ

<image>As shown in the figure, AC = AD, ∠BAC = ∠BAD, and point E is on AB. (1) Can you identify a pair of congruent triangles; (2) Please name a pair of congruent triangles and explain the reason.

\text{△ACB \cong △ADB

<image>As shown in the figure, in triangle $$\triangle ABC$$, $$AB=AC$$, $$BC=BD$$, and $$AD=DE=EB$$, what is the measure of angle $$\angle A$$ in degrees?

45

<image>(2016 Spring • Changle City Midterm) As shown in the figure, in the equilateral triangle ABC, point D and E are the midpoints of AB and AC respectively. Then the measure of ∠DEC is (__) degrees.

120°

<image>ABC takes the form of a sector. Side DE extends into an equilateral triangle outside the rectangle. Side GH is forming an equilateral triangle. What is the length of IG in shape FDGIH?

11

<image>In the , two monkeys are at point B which is 10 meters high on a tree. To compete for the fruit at point A by the pond, one monkey climbs down the tree and runs to point A, which is 20 meters away from point C. The other monkey climbs to the top of the tree at point D and then jumps directly to point A, the distance being measured in a straight line. If both monkeys travel the same distance, find the height of the tree.

15

<image>If you use a 145 cm long wooden strip to make a frame for your certificate (with a total loss of 4 cm at the joints). (1) Is this strip long enough? (2) To cover the certificate with a piece of glass, at least how large should the glass be?

Enough

<image>As shown in the , in triangle ABC, ∠ACB = 90°, ∠A = 20°. Triangle ABC is rotated around point C in the counterclockwise direction by an angle α to the position of triangle A′B′C′, where A′ and B′ are the corresponding points of A and B, respectively, B is on A′B′, and CA′ intersects AB at D. Then, the degree measure of ∠BDC is ____ degrees.

60

<image>ABCD is a parallelogram. FEG takes the form of a sector. GEH is an isosceles triangle where GE = EH. I need to know the length of the base GH in the isosceles triangle GEH.

22*√(2)

<image>As shown in the figure, in the square $ABCD$, $AB = 3$, point $E$ is on the side $CD$ such that $CD = 3DE$. Fold $\triangle ADE$ along $AE$ to $\triangle AFE$. Extend $EF$ to intersect $BC$ at point $G$. Connect $AG$ and $CF$. The following conclusions are given: $①$ Point $G$ is the midpoint of $BC$; $②$ $FG = FC$; $③$ $\angle GAE = 45^{\circ}$; $④$ $\angle FCG = 60^{\circ}$. Among these, the correct statements are ____________.

①③

<image>DCE is an isosceles triangle where the length of DC equals the length of CE. Square ECFG is inscribed with a circle. In the shape ECFG, what is the measurement of EG?

22

<image>In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is $X$, in centimeters?

5

<image> As shown in the figure, the slope of _$$AB$$ is _$$1$$:$$3.$$ If the slope length _$$AB$$ is 10m, then the height of the slope _$$BC$$ is ______$$m$$.

\sqrt{10

<image>【Problem】As shown in the figure, AD⊥BC, CE⊥BC, CH⊥AB, BG⊥AC. In △ABC, the altitude from the side BC is (_____) Choices: (A) Line segment CE (B) Line segment CH (C) Line segment AD (D) Line segment BG

Line segment AD

<image>As shown in the , line AB intersects with line CD at point O. EO is perpendicular to CD, and ∠BOE is 48°. The degree measure of ∠AOD is (___).

138

<image>ABC takes on a sector shape. GHI shapes into an equilateral triangle. What is the overall perimeter of FEGIH?

22*√(2) + 33

<image>As shown in the , in triangle $$\triangle ABC$$, $$∠1=100^{\circ}$$, $$∠C=80^{\circ}$$, $$∠2= \dfrac{1}{2}∠3$$, and $$BE$$ bisects $$∠ABC$$. Find the measure of $$∠4$$.

45^{\circ

<image>As shown in the figure, the radius of circle $$⊙O$$ is $$10$$. The length of chord $$AB$$ is $$12$$. The line segment $$OD$$ is perpendicular to $$AB$$, intersecting $$AB$$ at point $$D$$ and intersecting circle $$⊙O$$ at point $$C$$. Then, $$OD =$$ ______ , $$CD =$$ ______ .

8

<image>(Spring 2015, Final Exam in Shengzhou City) As shown in the figure, in the parallelogram ▱ABCD, AB = 8 cm, AD = 5 cm, the bisector of ∠BAD intersects CD at point E, and the bisector of ∠ABC intersects CD at point F. Then, the length of the segment EF is ____ cm.

2

<image>As shown in the figure, the distance between two adjacent points in the horizontal and vertical directions is m. If the area of quadrilateral ABCD is 23, then the area of pentagon EFGHI is ____.

28

<image> As shown in the figure, a rectangular piece of paper ABCD is folded along EF, and points C and D fall on positions C′ and D′ respectively. EC′ intersects AD at point G. Given that ∠EFG=58°, then ∠BEG=(___).

64

<image>Uncle Wang wants to pave the train station square with colored tiles (as shown in the image), leaving a square flower bed with a perimeter of 28 meters. How many square meters of colored tiles are needed?

4001

<image>ABC forms a sector. CBD is an isosceles triangle where the sides CB and BD are identical in length. DBE conforms a sector. In the sector DBE, what is the measurement of arc DE?

99*π/2

<image>(2015 Spring•Tongliao Final Exam) As shown in the figure, if AB∥DC, ∠1=39°, and ∠C and ∠D are complementary, then ∠D=____, ∠B=____.

39°, 129°

<image> A park is planning to build a rectangular lawn, with a length of 30 meters and a width of 20 meters. A cross-path as shown in the figure will be constructed on the lawn, with the width of the path being x meters. Answer the following questions: (1) What is the area of the cross-path in square meters? (2) If the width of the path is 2 meters, what is the area of the lawn (the shaded part) in square meters?

50x - x^2

<image>As shown in the figure, in $$\triangle ABC$$, $$∠ABC = 50º$$, $$∠ACB = 75º$$, point $$O$$ is the incenter of $$\triangle ABC$$, then the measure of $$∠BOC$$ is ________ .

117.5^\circ

<image>If the length of the AD side is 17, the area of the ABCD parallelogram is 114, the length of the BE side is 7, the degree of the FEB angle is 70, the adjacent angles BEA and FEB are complementary, the length of the AG side is 14 and the diagonal of the AGHE rectangle is 19, compute the degree of the DAB angle. Round computations to 2 decimal places.

90

<image>ABC forms a sector. CBD is an isosceles triangle, having CB and BD of the same length. DBEF conforms a square. Side GH forms an equilateral triangle inside the rectangle. How do you find the perimeter of EBGIH?

352

<image>ABCD conforms a square. Side GH transitioning into an equilateral triangle outside. Can you calculate the area of the entire shape FEGIH?

400*√(3) + 3000

<image>As shown in , to bend the angle iron (Figure ①) into a 145° steel frame (Figure ②), the notch cut out on the angle iron (dotted line in Figure ①) should be (__ degrees).

35

<image>ABC takes the form of a sector. CBDE is geometrically a parallelogram. EDF takes the form of a sector. Side GH converting into a semi-circle outside. Can you calculate the area of the entire shape FDGH?

8281*π/8 + 4641

<image>As shown in the figure, all squares have their centers at the origin, and all sides are parallel to either the $$x$$-axis or the $$y$$-axis. From the inside out, their side lengths are $$2$$, $$4$$, $$6$$, $$8$$, $$...$$, and their vertices are denoted sequentially as $$A_{1}$$, $$A_{2}$$, $$A_{3}$$, $$A_{4}$$, $$...$$. What are the coordinates of vertex $$A_{2017}$$?

(-505,-505)

<image>DCE is in the form of a sector. ECFG conforms a parallelogram. Side HI arcs into a semi-circle inside the rectangle. In shape GFHI, what is the length of HI?

48*π

<image>Please find the areas of the following three figures in the grid below. Each small square represents 1 square centimeter, and any incomplete square should be counted as half a square. (1) What is the approximate area of the butterfly? (2) What is the approximate area of the dragonfly? (3) What is the approximate area of the maple leaf?

20

<image>ABCD conforms a square. FEGH is geometrically a square. How much is the area of the entire shape FEGH?

81

<image>In △ABC, AB = AC, and DE is the perpendicular bisector of AB, intersecting AB and AC at D and E respectively. If ∠A = 40°, find the measure of ∠EBC

30

<image>As shown in the figure, through the midpoint E of the edge AD of the tetrahedron ABCD, a section EFGH is made parallel to edges AB and CD. If AB = 4 and CD = 6, then the perimeter of the section EFGH is _______________.

10

<image>Find the area of each of the following shapes.

25cm^2

<image>As shown in the figure, in △ABC, BD is the angle bisector of ∠ABC, DE ∥ BC, intersecting AB at E. Given that ∠A = 60° and ∠BDC = 95°, what is the measure of ∠BED? (_ )

110

<image>As shown in the figure, line segment $$OC$$ is a ray inside angle $$∠AOB$$, and $$OM$$ and $$ON$$ bisect angles $$∠AOC$$ and $$∠BOC$$, respectively. (1) If $$∠AOC={120}^{0}$$ and $$∠BOC={30}^{0}$$, find the measure of $$∠MON$$; (2) If $$∠AOC=α$$ and $$∠BOC=β$$, express $$∠MON$$ using algebraic expressions involving $$α$$ and $$β$$, and directly state the quantitative relationship between $$∠AOB$$ and $$∠MON$$.

75^\circ

<image>Cut a square piece of colored paper with a side length of 6 cm into a "Z" shape, as shown in the image. Find the area of the "Z" shape.

14.84

<image> As shown in the figure, the exterior angle of △ADC is ____.

∠ADB, ∠EAC

<image>ACD is a sector. DCE is an isosceles triangle with DC and CE being equal in length. What is the perimeter length of DCE?

48 + 48*√(2)

<image>ABCD is a parallelogram. DCEF conforms a square. FEGH conforms a square. What is the side length GE in the shape FEGH?

52

<image>As shown in the figure, a student uses a right triangle with a 60° angle to estimate the height of the school flagpole AB. He places the horizontal leg of the 60° angle on a 1.5-meter high stand CD, aligning the hypotenuse of the triangle with the top of the flagpole. He measures the distance from D to B as 5 meters. The height of the flagpole AB is approximately ______ meters (accurate to 1 meter, $\frac{} = 1.73$).

10

<image>ABCD is identified as a parallelogram. FEG is in the form of a sector. GEHI is identified as a parallelogram. How much area does the shape GEHI cover?

√(3)/2

<image>(2010 Tianhe District Mock Exam) As shown in the figure, a square ABCD with a side length of 2 cm is cut along its diagonal AC, and then △ABC is translated along BC to obtain △A′B′C′. If the overlapping area of the two triangles is denoted as S, the maximum value of S is ___ cm^{2}.

1

<image>If the brown shape is a rectangle where an equilateral triangle has been removed from one side of it, the area of the brown shape is 114, the area of the lime sector is 56.52 and the angle $\theta$ and the adjacent 30 degree angle are complementary, compute the length of the side of the brown shape marked with question mark. Assume $\pi=3.14$. Round computations to 2 decimal places.

15.48

<image>(2014 Autumn • Ouhai District Midterm Exam) As shown in Figure 1, this is a diagrammatic representation of the famous ancient Chinese "Zhao Shuang's Inscribed Square Diagram". It is formed by four congruent right-angled triangles. If the shorter leg BC = 2.5, and the longer legs of the four right triangles are each extended outward by an equal amount, resulting in the "Mathematical Windmill" shown in Figure 2, and if the perimeter of △BCD is 15, then the outer perimeter of this windmill is ____.

38

<image>As shown in the figure, $$∠1 = ∠3$$, and $$∠1 + ∠2 = 180^{\circ}$$. $$(1)$$ Determine the positional relationship between $$GF$$ and $$CB$$, and explain the reason;___ $$(2)$$ If $$BF ⊥ AC$$ and $$∠2 = 150^{\circ}$$, find the degree measure of $$∠AFG$$.$$__

60°

<image>A math extracurricular group measured the height of the sculpture CD at Jinhu Plaza. They measured the angle of elevation to point D at the top of the sculpture from point A on the ground to be 30°. Moving 18 meters forward towards the base of the sculpture to point B, they measured the angle of elevation to be 45° (as shown in the image). Find the height of the sculpture CD. (accurate to 0.01 meters)

24.59

<image>EFG forms a triangle with all sides equal. GEH takes on a sector shape. Square HEIJ is inscribed with a circle. What is the total boundary length of circle?

π

<image>As shown in the figure, points A, B, and C are on the circumference of circle O. If ∠ABC = 40°, then the degree measure of ∠AOC is _________. __

80

<image>DEF creates an equilateral triangle. EFGH forms a square. What would the area of the entire shape EHI be?

3/2