Geometry and measures worksheet — GCSE Higher

1.A, B, C and D are points on a circle, and ABCD is a cyclic quadrilateral with the vertices in that order around the circle. Angle ABC = 108° and angle ADC = 72°. Which circle theorem is the reason that angle ABC + angle ADC = 180°?

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

3.A quadrilateral has all four sides equal in length and all four interior angles equal to 90°. Its diagonals are equal in length and bisect each other at right angles. Which quadrilateral matches this description exactly?

4.A hiker starts by walking on a bearing of 245°. She turns clockwise through 50°, then turns clockwise through a further 15°, and continues walking in a straight line. What bearing is she now walking on?

5.In triangle ABC and triangle DEF, the angle at A equals the angle at D, and AB ÷ DE = AC ÷ DF. Write down what this proves about the two triangles.

6.A garden flower bed is in the shape of a trapezium. Its two parallel edges are 3.5 m and 6.5 m long, and the perpendicular distance between them is 4 m. Bedding plants cost £2.50 per square metre to cover the bed. Work out the total cost of the bedding plants.

7.A surveyor marks two fixed points A and B, with position vectors OA = a and OB = b (in km) from a base station O. A relay mast P is to be placed on the line AB such that AP : PB = 3 : 2. Express the vector OP in terms of a and b.

8.ABCD is a cyclic quadrilateral, with its vertices in that order around the circle. Angle DAB = (2x + 10)° and angle DCB = (3x − 5)°. Work out the size of angle DAB.

9.An isosceles trapezium has exactly one pair of parallel sides, and its two non-parallel sides are equal in length. How many lines of symmetry does it have?

10.A cylindrical candle and a cone-shaped candle have the same base radius and the same height. State the cone's volume as a fraction of the cylinder's volume.

11.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?

12.A wooden cube has edges of length 4 cm. Work out the total surface area of the cube.

13.Work out the y-coordinate of the midpoint of the line segment joining (2, 4) and (8, 10).

14.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.

15.In triangle OAB, OA = a and OB = b. M is the midpoint of OA, and N is the midpoint of OB. Express the vector MN in terms of a and b.