Geometry and measures worksheet — GCSE Higher

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

2.A parallelogram-shaped tile has two sides of 6 cm and 9 cm, with an angle of 60° between them. Work out the area of the tile. Give your answer to 1 decimal place.

3.A trapezium has vertices (2, 1), (6, 1), (5, 4) and (3, 4). It is rotated 180° about the vertex (2, 1). Work out the number of points on the trapezium — including its vertices, edges and interior — that are invariant under this rotation.

4.From an external point P, two tangents PA and PB touch a circle with centre O at A and B. Angle AOB = 112°. Work out the size of angle APB.

5.A straight line touches the edge of a circle at exactly one point and does not cross into the circle at all. Write down the term for this line.

6.Point P is translated by the vector (5, −2), and the image is then rotated 90° clockwise about the origin to give the point (3, 9). Work out the coordinates of P.

7.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.

8.A trapezium-shaped allotment plot has two parallel sides that face each other, and two sloping sides that are equal in length (an isosceles trapezium). One of the angles next to the shorter parallel side is 118°. Work out the angle next to the other end of the same shorter parallel side.

9.A water tank is a cube with edges of length 2 m. Work out how many cubic centimetres the tank holds when it is full.

10.ABCD is a cyclic quadrilateral, with its vertices in that order around the circle. Angle DAB = 108°. Work out the size of angle BCD.

11.In triangle ABC, angle ABC = 90° and angle BAC = 30°. The hypotenuse AC = 12 cm. Work out the exact length of AB.

12.In geometry, which of the following best describes a 'plane'?

13.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.

14.In triangle ABC and triangle DEF, AB = DE, AC = DF, and the angle at A equals the angle at D. Write down the congruence condition that proves the two triangles are congruent.

15.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.