Geometry and measures worksheet — GCSE Higher

1.A roof truss has two horizontal parallel rafters, one above the other. A straight strut crosses both rafters. Where the strut crosses the lower rafter, the angle above the rafter and to the left of the strut is 65°. Work out the size of the angle above the upper rafter and to the right of the strut, where the strut crosses it.

2.The diagram shows a solid made from two cuboids joined together. Which shape is the front elevation of this solid?

3.OABC is a parallelogram, with OA = a and OC = c. M is the midpoint of OC, and N is the point on AC such that AN is twice NC. By finding the vectors MN and MB, show that M, N and B are collinear, and give the scalar k such that MN = k × MB.

4.OABC is a parallelogram, with OA = a and OC = c. M is the midpoint of AB. Express the vector OM in terms of a and c.

5.A solid metal sphere has a radius of 5 cm. It is melted down and recast into a solid cylinder with the same radius, 5 cm. Work out the height of the cylinder, so that its volume equals the volume of the sphere. Use π = 3.14 and give your answer correct to 1 decimal place.

6.A trundle wheel has a diameter of 0.5 m. It is rolled along the ground and makes 20 complete turns. Using π = 3.14, work out the total distance rolled, in metres.

7.A triangle has vertices X, Y and Z, with a right angle at Y. Side XY is twice as long as side YZ. Which of the following correctly describes this triangle?

9.The midpoint of the line segment AB is (3, 5). A is the point (1, 3). Work out the coordinates of B.

10.The point A(3, 2) is translated by the vector (−5, 4) to give point B. Work out the coordinates of B.

11.m is the column vector with top number 3 and bottom number −4. Which of these column vectors is a scalar multiple of m?

12.A cylinder has a radius of 5 cm. Its volume is 471 cm³. Using π = 3.14, work out the height of the cylinder.

13.In triangle ABC, AB = 15 cm, angle BAC = 38° and the area of the triangle is 61 cm². Work out the length of BC. Give your answer to 1 decimal place.

14.Two metal window frames are checked before installation. Frame A has edges of 40 cm and 65 cm, with a marked angle of 35° at the far end of the 65 cm edge, away from where the two given edges meet. Frame B has matching edges of 40 cm and 65 cm, with a matching 35° angle in the same position. The fitter wants to know if these measurements alone guarantee the two frames are congruent. What should the fitter be told?

15.A kitchen scale is marked in equal divisions of 20 g. The pointer rests three divisions past the 400 g mark. Work out the reading on the scale.