Geometry and measures worksheet — GCSE Higher

1.A triangle has vertices X, Y and Z, with a right angle at Y. Side XY is twice as long as side YZ. Which of the following correctly describes this triangle?

2.Two similar triangles have lengths in the ratio 2 : 5. The sides of the smaller triangle are 4 cm, 6 cm and 8 cm. Work out the length of the longest side of the larger triangle.

3.A regular hexagon is divided into six identical triangles by joining its centre to each of the six vertices. Work out the size of the angle of one of these triangles at the centre of the hexagon.

4.Triangle D has vertices (3, 1), (3, 5) and (6, 1). It is reflected in the line x = 3. Write down the number of vertices of the triangle that are invariant under this reflection.

5.The midpoint of the line segment AB is (3, 5). A is the point (1, 3). Work out the coordinates of B.

6.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?

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

8.A trapezium-shaped garden bed has parallel sides of 2.5 m and 4.5 m, and a perpendicular width of 3 m. A gardener wants to know the area of the bed before ordering topsoil. Work out the area of the garden bed.

9.Two sides of a triangle are 20 cm and 8 cm long. Write down which one of these lengths is possible for the third side.

10.Two points on a circle divide its circumference into two arcs of different lengths. Write down the name given to the shorter of the two arcs.

11.A quadrilateral has exactly two lines of symmetry, its two pairs of opposite angles are equal, and its diagonals are not equal in length. Write down the name of this quadrilateral.

12.Two triangular offcuts of wood, PQR and STU, are cut for a construction project. PQ = 8 cm, QR = 6 cm and angle PQR = 90°. ST = 8 cm, TU = 6 cm and angle STU = 90°. A carpenter wants to check the two pieces are identical in shape and size before using them as a matching pair. Using only the measurements given, and without working out any further lengths, which condition proves that triangle PQR is congruent to triangle STU?

13.Square B has vertices (1, 1), (4, 1), (4, 4) and (1, 4). It is rotated 90° clockwise about the origin, and the image is then translated by the vector (2, −1). Work out the coordinates of the image of (4, 1).

14.A fan-shaped rug is in the shape of a sector of a circle with radius 4 m and angle 180°. It costs £4 per square metre to make. Using π = 3.14, work out the total cost of the rug.

15.A triangular plot of land has sides AB = 13 m and AC = 10 m, with angle BAC = 72° between them. Work out the perimeter of the plot. Give your answer to 1 decimal place.