Math practise (geometry) with solutions

Geometry, mensuration practice questions 1 Points A, B, C and D lie on a circle. AC is a diameter of the circle. AEC and BEDF are straight lines. AF // BC. AD = AC. Angle ADB = 36° (a) Show that triangle AED and triangle BEC are similar [3] (b) Find (i) angle ACB [Ans: 36°] [1] (ii) angle BAD [Ans: 72°] [2] (iii) angle CAD [Ans: 18°] [2] (iv) angle DAF [Ans: 18°] [1] (c) Hence, explain why AD = DF. [Ans: angle DFA = angle DAF = 18°] [2]

2 The diagram below shows a regular hexagon with sides 10 cm. (a) (i) Find the interior angle of the regular hexagon. [Ans: 120°] [2] (ii) Hence, find the area of the regular hexagon. [Ans: 260 cm2] [3] (b) In the diagram, a hexagonal pyramid with ON as the height and sides of 10 cm. M is the midpoint of AB. AN = 13 cm. (i) Calculate the volume of the pyramid. [Ans: 720 cm3] [4] (ii) Calculate the total surface area of the pyramid. [Ans: 440 cm2] [3]

3 A (0,4) and B lie on the y-axis. D lies on the x-axis. A is the midpoint of OB. OB : OD = 2 : 3. (a) Find the equation of AC. [Ans: y = 2/5x + 4 or 5y = 2x + 20] [2] (b) Find the length BC. [Ans: 15.1 units] [2] (c) Calculate the area of triangle ABC. [Ans: 30 units2] [2] (d) Find tan angle ODB. [Ans: 33.7°] [2] (e) Calculate the area of triangle BOD. [Ans: 48 units2] [2] 4 A nonagon (9-sided polygon) has six (x + 2)° and three (2x)° interior angles. Find the largest interior angle. [Ans: 208°] [4]

5 A circle with centre O has a radius of 8 cm overlaps with rectangle PQRS. The length of PQRS is 15 cm and its perimeter is 40 cm. (a) Calculate the shaded area. [Ans: 18.1 cm2] [5] (b) Calculate the perimeter of the shaded region. [Ans: 44.4 cm] [4] 6 A sports company manager wanted to make tennis ball containers. Each tennis ball has a radii of 3.35 cm. He wanted to fit exactly 4 tennis balls into two different boxes. Box A is a closed cuboid while Box B is a closed cylinder. Both boxes have the same height. Box A Box B The cost per cm2 of plastic to make the box is $0.60. Which box should the manager choose if he only puts aside a budget of $400. Show all your workings. [Ans: Box A costs $484.81, Box B costs $380.77. He should choose Box B.] [7]

7 AB and CD are sides of the two buildings. The angle of elevation of C from B is 42°. The angle of depression of C from A is 38°. The distance between the two buildings is 56 m. Find the height of the taller building. [Ans: 94.2 m] [5] 8 Diagram I show a kitchen sink. The sink is a trapezoidal prism with height 15 cm. The prism (sink) is removed from the cuboid (table). Diagram II show the plane view of the sink. All dimensions are in centimetres. Calculate the volume of the remaining solid. [Ans: 235295 cm3] [4]

9 OAB is a triangle. 𝑂𝐴⃗⃗⃗⃗⃗ = a and 𝑂𝐵⃗⃗⃗⃗⃗ = b. OQ : OB = 3 : 5. P is the midpoint of A and B. (a) Express your answers in terms of a and b. Find (i) 𝐴𝐵⃗⃗⃗⃗⃗ [Ans: b - a] [1] (ii) 𝑂𝑃⃗⃗⃗⃗⃗ [Ans: ½ a + ½ b] [2] (b) Explain why ABQX is a trapezium. [Ans: 𝐴𝐵⃗⃗⃗⃗⃗ = 3/10 𝑋𝑄⃗⃗⃗⃗⃗ ] [3] (c) Find (i) area of triangle OPB : area of triangle OAB. [Ans: 1 : 2] [1] (ii) area of PBQX : area of triangle OAB. [Ans: 2 : 5] [2]

10 (a) PQRS and TUVS are congruent quadrilaterals. Angle PQR = 165° and angle UVS = 67°. Angle QPS = angle UTS = 90°. PS = 5.6 cm, TU = 4.9 cm and VS = 7.8 cm. Find (i) angle TUV [Ans: 165°] [1] (ii) angle PSV [Ans: 76°] [2] (iii) TR [Ans: 2.2 cm] [2] (b) In the diagram, AB = AQ, PA = QS and AB is parallel to QR. QR and AC are produced to meet at S.

(i) Name the triangle that is congruent to triangle P AB. [Ans: Triangle SAQ] [1] (ii) Show that triangle P AC is similar to triangle SRC. [Ans: AA Test] [2] Given that AP = 9 cm, AC = 3 cm and RS = 5 cm, find the length of (iii) CR [Ans: 1.67 cm] [2] (iv) AB [Ans: 3 cm] [4] (c) A, B and C are similar cones. The ratio volume of cone A : volume of cone B : volume of cone C = 1 : 3 : 8. (2022 O level Paper 2) (i) Find the ratio height of cone A : height of cone C. [Ans: 1 : 2] [1] (ii) Find the surface area of cone B as a percentage of the surface area of cone C. [Ans: 52.0 %] [2]