Geometry and measures worksheet — GCSE Higher

1.A path goes from A(1, 1) to B(1, 5), then from B to C(6, 5). Work out the total length of the path from A to C.

2.Shape S has an area of 5 cm². Shape S is enlarged by a scale factor of −3 to give shape T. Work out the area of shape T.

3.A right-angled triangle has a hypotenuse of 10 cm. One of its other angles is 45°. Work out the exact length of one of the two shorter sides.

4.Shape S is enlarged by a scale factor of −3 about a fixed centre. Which statement correctly describes the image compared to the original shape?

5.Triangle PQR has angle P = 90°. The side PQ = 9 cm and the hypotenuse QR = 15 cm. Work out the length of PR.

6.In a construction, a point P is marked at the foot of the perpendicular drawn from point X to a straight line. Which statement correctly compares the distance XP with the distance from X to any other point on the line?

7.A hexagonal prism has 8 faces and 12 vertices. Using F + V − E = 2, work out how many edges it has.

8.In triangle OAB, OA = a and OB = b. P is the point on OA such that OP = (1/3)a, and Q is the point on OB such that OQ = (1/3)b. Express the vector PQ in terms of a and b.

9.A roof has two triangular sections, P and Q. Section P has a sloping edge of 3.6 m and a base edge of 2.4 m, with an angle of 70° between them. Section Q has a sloping edge of 3.6 m and a base edge of 2.4 m, with an angle of 70° between them, arranged the same way round. A roofer wants to cut identical triangular felt panels for both sections without measuring Section Q separately. Which condition confirms that the two sections are congruent?

10.A cuboid has a length, width and height that are all different from each other. How many planes of symmetry does it have?

11.In triangle ABC, AB = 7.4 cm, AC = 5.9 cm and angle BAC = 68°. Work out the length of BC. Give your answer to 1 decimal place.

12.A construction for the perpendicular bisector of AB begins by opening a pair of compasses to more than half the length of AB, then drawing an arc centred at A. What is the next step?

13.A hexagon has interior angles of 130°, 142°, 125°, 150°, 135° and x°. Work out the value of x.

14.Triangle ABC and triangle DEF are similar. The area of triangle ABC is 36 cm² and the area of triangle DEF is 100 cm². Work out the ratio of the lengths of triangle ABC to the lengths of triangle DEF, in its simplest form.

15.Two spheres are similar solids. Their radii are in the ratio 3 : 4. Write down the ratio of their surface areas.