Math topic中文

Pythagoras

use of Pythagoras' theorem

Practice questions

  1. 1In the diagram, there are two right-angled triangles ΔABD and ΔCEB such that AD = 12 cm, BD = 15 cm and BE = 20 cm. Find…
  2. 2In triangle ABC, ∠ABC=90∘\angle ABC = 90^\circ, BX=2XCBX = 2XC and sin⁡∠AXC=35\sin \angle AXC = \dfrac{3}{5}. Express as a fraction in its…
  3. 3In the diagram, POR is a quadrant of a circle with radius 6 cm. OR and PQ are parallel. QR is an arc of a circle with ce…
  4. 4Triangle ABC is a right-angled triangle. Given that two of its lengths are 11 cm and 5 cm, find two possible lengths for…
  5. 5The diagram below shows a gift souvenir formed by joining the plane face of a solid hemisphere of radius 3 cm and a soli…
  6. 6In the diagram, PQRS is a vertical rectangular wall of height 2.5 m and length 7.5 m. T is a point on level ground from…
  7. 7三角形的三边长分别为aa、bb、cc，且满足等式：(a+b)2−c2=2ab(a+b)^{2}-c^{2}=2ab，则此三角形是()() A. 锐角三角形 B. 直角三角形 C. 钝角三角形 D. 等腰三角形
  8. 8圆CC的半径为55，圆心在xx轴的负半轴上，且被直线3x+4y+4=03x+4y+4=0截得的弦长为66，则圆CC的方程为(      )\left(\;\;\;\right) A. x2+y2−2x−3=0x^{2}+y^{2}-2x-3=0…
  9. 9(1)二进制数学1101101(2)1101101_{(2)}转化为十进制数为___. (2)若实数xx，yy满足约束条件{x⩾0y⩽x2x+y−9⩽0\left\{\begin{array}{l}x\geqslant 0\\ y\leqslant x\\ 2x+y-9\leqslant 0\end{array}\right.…
  10. 10An industrial mixer is made up of an open cylinder connected to a cone. The height of the cylinder and the cone is 1.2 m…
  11. 11在平面直角坐标系中，点OO为坐标原点，直线y=kx+k(k>0)y=kx+k(k > 0)与xx轴交于点AA，与yy轴交于点BB，过点AA的抛物线y=x2+bx+cy=x^{2}+bx+c与xx轴交于另一点P(1)P(1)…
  12. 12In the diagram, ABCDEFGH is a cuboid with dimensions 8 cm by 6 cm by 15 cm. V is the centre of the rectangular base. Sho…
  13. 13已知圆的方程(x−2)2+y2=1{{(x-2)}^{2}}+{{y}^{2}}=1，过圆外一点P(3,4)P\left(3,4\right) 作一条直线与圆交于A,BA,B两点，那么PA→⋅PB→=\overrightarrow{PA}\cdot \overrightarrow{PB}=…
  14. 14The diagram shows a right circular cone cut from a solid steel sphere. Point O is the centre of the sphere with radius 1…
  15. 15在△ABC\triangle ABC中，∠C=90°∠C=90°，tan⁡A=43\tan A=\dfrac{4}{3}，AB=20AB=20，则AC=AC=___.
  16. 16Triangle PQR is a right-angled triangle with PQ = 10 cm, QR = 6 cm and ∠PRQ = 90°. M is a point on PR such that MR = 5 P…
  17. 17下列四组数中，是勾股数的是()(\quad) A. 2.52.5、66、6.56.5 B. 3232、4242、5252 C. 11、22、5\sqrt{5} D. 55、1212、1313…
  18. 18The diagram shows a semi-circle BAC, with diameter BC = 34 cm. AD and BCD are straight lines. Given that cos⁡∠ABC=817\cos\angle ABC = \dfrac{8}{17}…
  19. 19下列各组数中，是勾股数的是()(\quad) A. 11，11，2\sqrt{2} B. 2.52.5，4.54.5，44 C. 66，77，88 D. 99，1212，1515…
  20. 20《九章算术⋅\boldsymbol{⋅}商功》：“斜解立方，得两塹堵，斜解塹堵，其一为阳马，一为鳖臑．阳马居二，鳖臑居一，不易之率也．合两鳖臑三而一，验之以基，其形露矣．”文中“阳马”是底面为长方形且有一条侧棱与底面垂直的四棱锥．在阳…
  21. 21从点P(3,−1)P(3,-1)向圆x2+y2−4y−5=0x^{2}+y^{2}-4y-5=0作切线PT(TPT(T为切点))，则∣PT∣|PT|等于()(\quad) A. 5                                           5\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: …
  22. 22Figure I shows a piece of wood is in the form of a right circular cone with a base diameter of 18 cm. The curved surface…
  23. 23勾股定理最早出现在商高的《周髀算经》：“勾广三，股修四，经隅五”．观察下列勾股数：33，44，55；55，1212，1313；77，2424，2525；…，这类勾股数的特点是：勾为奇数，弦与股相…
  24. 24已知圆C1\mathbf{C}_{\mathbf{1}}：x2+y2-6x-6=0\mathbf{x}^{\mathbf{2}}\mathbf{\text{+}}\mathbf{y}^{\mathbf{2}}\mathbf{\text{-}6}\mathbf{x\text{-}6\text{=}0}…
  25. 25The diagram shows two open containers, a right cylinder and an inverted frustum. The right cylinder has a base radius of…
  26. 26The diagram shows a cardboard in the shape of a major sector, centre O and radius 15 cm. The total area of the major sec…
  27. 27直线y=2xy=2x与圆C:(x−2)2+(y+1)2=14C:(x-2{)}^{2}+(y+1{)}^{2}=14交于AA，BB两点，则△ACB{\rm\triangle}ACB的面积等于().(\quad). A. 352\dfrac{3\sqrt[]{5}}{2}…
  28. 28ABCD is a parallelogram. A is the point (−6,−2)(-6,-2) and B is the point (−4,−7)(-4,-7). BC→=(8−2)\overrightarrow{BC} = \begin{pmatrix}8\\-2\end{pmatrix}…
  29. 29The diagram shows a solid formed from a cone, a cylinder and a hemisphere. The cone has base radius 1.5x1.5x cm and height…
  30. 30In the diagram, BDCE is a straight line. BD = 4 cm, AC = 10 cm and AB = AD. Given that the area of triangle ABD is 16 cm…
  31. 31The diagram shows a cylindrical container used to dispense coffee in a hotel. Calculate the volume of the cylinder. Answ…
  32. 32直线ll：kx−y+1=0kx-y+1=0被圆x2+y2−4y=0x^{2}+y^{2}-4y=0截得的最短弦长为()() A. 232 \sqrt {3} B. 33 C. 222 \sqrt {2} D. 22
  33. 33In the diagram, a wooden ramp is shown where ∠BCF=90∘\angle BCF = 90^\circ and ∠CBF=35∘\angle CBF = 35^\circ. ABCD, CDEF and ABFE ar…
  34. 34The equation of the line l1l_1 is x+4y=20x + 4y = 20. The line cuts the x-axis at point A and the y-axis at point B. Another…
  35. 35一个正三棱锥的底面边长为66，侧棱长为15 \sqrt {15}，那么这个正三棱锥的体积是 ___ ．
  36. 36The figure shows two rods PA and PB which are hinged at P. The rods slide such that P, A and B are on the same vertical…
  37. 37O is the centre of the equilateral triangle. The triangle ABC forms the base of a pyramid. The vertex, X is vertically a…
  38. 38AC is a diameter of a circle with centre O. AC=2rAC = \sqrt{2}r and AB=BCAB = BC. COB is a semi-circular arc, centre P. What p…
  39. 39P is the point (-3, 1) and Q is the point (3, 3). Find the length of the PQ. The point R is such that PQR forms an isosc…
  40. 40某圆锥的侧面展开图扇形的弧长为8π8π，扇形的半径为55，则圆锥的体积为()(\quad) A. 25π25π B. 7575 C. 55π5\sqrt{5}π D. 16π16π
  41. 41已知直线ll：kx+y−2=0(k∈R)kx+y-2=0(k∈R)是圆CC：x2+y2−6x+2y+6=0x ^{2} +y ^{2} -6x+2y+6=0的一条对称轴，过点A(0,k)A(0 , k)作圆CC的一条切线，切点为BB，则线段ABAB的长度为…
  42. 42The diagram below shows an indoor adventure park in the shape of a trapezoidal prism with a square base ABCD. The indoor…
  43. 43Rectangle ABCDABCD is inscribed in a semicircle whose diameter is FE‾\overline{FE}, as shown in the figure. Given that DA=16DA = 16…
  44. 4422 P, Q and R are three points on level ground such that RPQ forms a right-angled triangle with PQ and RQ measuring 135…
  45. 45已知圆CC的圆心在直线y=x+1y=x+1上，且圆CC与xx轴相切，点P(−5,−2)P(-5,-2)在圆CC上，点Q(−4,−5)Q(-4,-5)在圆CC外． (1)求圆CC的方程； (2)若过点(−2,−4)(-2,-4)的…
  46. 46The diagram shows a lecture theatre in the shape of a prism with height 9 metres. The floor of the lecture theatre, ABCD…
  47. 47The diagram below shows three points, A, B and C on flat ground. AB = 80 m, AC = 100 m and BC = 90 m. C is due east of A…
  48. 48圆锥的母线长为55，圆锥高为33，则该圆锥的侧面积为___.(.(结果保留π)π)
  49. 49下列各组数据中，是勾股数的为()(\quad) A. 33，44，55 B. 13\dfrac{1}{3}，14\dfrac{1}{4}，15\dfrac{1}{5} C. 0.30.3，0.40.4，…
  50. 50在平面内用12根火柴棒首尾顺次相接搭成三角形，能搭成 个直角三角形，搭成 个等腰三角形．
  51. 51The position vector of A is (34)\begin{pmatrix}3\\4\end{pmatrix} and the position vector of B is (710)\begin{pmatrix}7\\10\end{pmatrix}…
  52. 52已知圆锥的高是12，底面圆的半径为5，则这个圆锥的侧面展开图的周长为 ．
  53. 53The diagram shows a solid ornament in the shape of a cylinder with an upright cone cut out at one end and a hemisphere a…
  54. 54In the pentagon ABCDE shown, angle BAE = 60° and angle BCD = 45°. AE is parallel to CD, E lies on CB extended and CE = D…
  55. 55在直径为4的圆内接矩形中，最大的面积是（ ） A. 4 B. 2 C. 6 D. 8
  56. 56若实数mm，nn满足∣m−3∣+n−4=0|m-3|+\sqrt{n-4}=0，且mm，nn恰好是Rt△ABC{\rm Rt}\triangle ABC的两条边长，则第三条边长为()(\quad) A. 33或44 B…
  57. 57A container is a prism with a cross section in the shape of a right-angled triangle. The container has a height of 12 cm…
  58. 58A chock is a block placed against a wheel or rounded object, to prevent it from moving. Jenny had to come up with a prot…
  59. 59Find an expression, in terms of rr, for the volume of the largest cube that can possibly fit inside a sphere of radius…
  60. 60The diagram shows two congruent isosceles triangles, PQS and RSQ. PQ and RS are parallel to the x-axis. Write down the c…