Geometry and measures worksheet — GCSE Higher

1.A shape is translated by the vector (5, −3), and the image is then translated by the vector (−5, 3). Which single transformation has the same overall effect as this combination, for every point?

2.A cyclist's journey from her house to the shop is represented by the column vector with top number 2 and bottom number 5, where each unit is 1 km east and 1 km north. She then cycles from the shop to the park, represented by the column vector with top number 4 and bottom number −3. What single column vector represents her journey from her house directly to the park?

3.A rectangular photograph has an area of 96 cm². One of its sides is 8 cm long. Work out the length of the other side.

4.A tangent to a circle with centre O touches the circle at point P, where OP = 5 cm. Point Q lies on the tangent so that PQ = 12 cm. Using the fact that a tangent is perpendicular to the radius at the point of contact, work out the length OQ.

5.Two traffic cones are geometrically similar. The smaller cone has height 6 m and the larger cone has height 9 m. Work out the ratio of the volume of the smaller cone to the volume of the larger cone, giving your answer in the form a : b in its simplest form.

6.Shape S is enlarged by scale factor −1, centre (0, 0). Which single transformation has the same overall effect as this enlargement, for every point?

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

8.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.

9.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.

10.A builder lays a low wall made from identical bricks. The diagram shows the plan of the wall's footprint on a grid, where each square is one brick. The wall is built up to a height of 4 bricks everywhere. Work out the total number of bricks used.

11.Postcard P is similar to postcard Q. The 6 cm side of P corresponds to the 15 cm side of Q, and the 8 cm side of P corresponds to a matching side of Q. Work out the length of that matching side of Q.

12.Two similar triangular tiles have lengths in the ratio 3 : 5. The longest side of the smaller tile is 9 cm. Work out the length of the longest side of the larger tile.

13.Two sides of a triangle are 9 cm and 15 cm long. Write down which one of these lengths is possible for the third side.

14.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.

15.PT is a tangent to a circle with centre O, touching the circle at T. C is a point on the circle such that C lies inside angle OTP (the right angle between the radius OT and the tangent PT), and OC is a radius. Angle TOC = 130°. Work out the size of angle PTC, the angle between the tangent PT and the chord TC.