Geometry and measures worksheet — GCSE Higher

1.Point Q has coordinates (3, 5). It is translated by the vector (1, −2), and the image is then enlarged by scale factor 3, centre the origin. Work out the coordinates of the final image of Q.

2.A designer enlarges a drawing of a model car for a poster. She first enlarges the drawing by a scale factor of 1.5, and then enlarges that result by a further scale factor of 2. On the original drawing, the position of a wheel relative to the front bumper is given by the column vector with top number 4 and bottom number −3, in centimetres. What is the corresponding column vector on the poster, in centimetres?

3.Triangle ABC is isosceles, with AB = AC. Angle B = 70°. Which reason correctly explains why angle C = 70°?

4.In a right-angled triangle, the side opposite angle θ is 6 cm and the hypotenuse is 10 cm. Work out the size of angle θ. Give your answer correct to 1 decimal place.

5.A hiker walks on a bearing of 065°. She then turns clockwise through 90° and continues walking in a straight line. What bearing is she now walking on?

6.Point B is at (3, 5). It is reflected in the line y = 2. Work out the coordinates of the image of point B.

7.A square-based pyramid has a square base whose diagonal is 14 cm, and each slant edge of the pyramid, from a base vertex to the apex, is 15 cm. Work out the height of the pyramid. Give your answer correct to 1 decimal place.

8.Work out the gradient of the line segment joining A(2, 3) and B(6, 11).

9.A garden water trough is a prism whose cross-section is a right-angled triangle with base 40 cm and height 30 cm. The trough is 120 cm long. Work out the volume of water needed to fill the trough completely.

10.P is the point (3, 4) and Q is the point (6, 0). Work out the distance of each point from the origin, and write down which point is closer to the origin.

11.A vertical flagpole casts a shadow 6 m long at the same time as a 1.2 m vertical post casts a shadow 1.8 m long. The flagpole and the post are both vertical, and the triangles formed by each object, its shadow, and the sun's ray are similar. Work out the height of the flagpole.

12.Shape S has a vertex at (9, 6). It is enlarged by a scale factor of 1/3, centre the origin. Work out the coordinates of the image of this vertex.

13.A tile is described as having two pairs of parallel sides, four equal sides, and diagonals that cross at right angles but are not equal in length. Give a reason why this tile cannot be a square.

14.Work out the exact value of sin 30° + cos 60°.

15.A model of a bridge is built to a scale where every length on the model is 1/240 of the matching length on the real bridge. A support beam on the model measures 3.6 cm. Work out the length of the real support beam, giving your answer in metres.