Geometry and measures worksheet — GCSE Higher

1.A scale drawing of a park uses a scale of 1 : 2000. A path is drawn 8.5 cm long on the drawing. A cyclist rides the length of the path and then rides straight back again along the same path. How far does the cyclist travel in total, in metres?

2.Quadrilateral PQRS has angle P = 92°, angle Q = 84° and angle R = 106°. Work out the size of angle S.

3.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?

4.Triangle DEF is isosceles, with DE = DF. Angle E = 58°. Give a reason why angle F = 58°.

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

6.A right-angled triangle has its two shorter sides equal to 8 cm and 15 cm. Work out the value of cos θ, where θ is the angle opposite the 8 cm side. Give your answer as a fraction.

8.A right-angled triangle has two sides of length 5 cm and 12 cm, and the angle between those two sides is 90°. Work out the length of the hypotenuse.

9.A circle has two straight radii drawn from the centre to the edge, together with the arc between them. Write down the name of the region enclosed by these two radii and the arc.

10.A vertical line passes through the point (5, 2). Write down the equation of this line.

11.In triangle XYZ, the side opposite vertex X is labelled using the standard lowercase-letter convention. Which label is correct?

12.A shop stacks identical storage crates in a display. The diagram shows the plan of the floor positions used, with one corner position marked. Every floor position is filled with crates stacked 3 high, except the marked corner position, which is left completely empty. Work out how many crates are used in total.

13.A ship's radar shows a lighthouse at the point (12, −4) on a grid measured in nautical miles. The ship is at (2, 5). The ship sails along the vector that takes it directly to the lighthouse, then sails along that same vector again. Work out the ship's final position.

14.A triangle has one unknown side or angle, marked x. In which one of these situations should the sine rule, rather than the cosine rule, be used to find x?

15.A pinhole camera is set up at a pitch-side stand, with its aperture at the point O = (2, 1) on a grid. Light from the corner F of a triangular flag, at the point (4, 2), passes through the aperture and forms an inverted image on the film behind it. The projection is an enlargement of scale factor −1.5 centred at O. Work out the coordinates of the image of corner F.