Geometry and measures worksheet — GCSE Higher

1.Triangle S has vertices (2, 2), (5, 2) and (2, 5). It is mapped onto triangle S′ with vertices (2, 5), (5, 5) and (2, 2). Which single composition of two transformations maps S onto S′?

2.Point A(1, 2) is enlarged to give image point A′(7, −4). The centre of enlargement lies on the x-axis. Work out the scale factor of the enlargement.

3.A cylindrical drum and a cuboid box are being wrapped for a display. A shop assistant wraps only the flat faces of each item, not any curved surface. How many flat faces in total does the assistant wrap across both items?

4.OAB is a triangle, with OA = a and OB = b. E lies on OA produced beyond A, such that A is the midpoint of OE. F lies on AB such that FB is twice AF. G is the midpoint of OB. By finding the vectors EF and EG, show that E, F and G are collinear, and give the scalar k such that EF = k × EG.

5.A conservatory has a roof panel shaped like a regular octagon. Work out the interior angle of the octagon, then use it to work out the size of the reflex angle at the same vertex, on the outside of the roof panel.

6.In triangle ABC the angle at C is 90° and the hypotenuse AB is 12 cm. H is the point on AB for which CH is perpendicular to AB, and AH = 3 cm. Work out the length of AC.

7.Two identical ladders lean against the same vertical wall from opposite sides, each making an angle of 58.2° with the ground. The two ladders and the ground form a triangle. Work out the size of the angle between the two ladders at the top, where they meet.

8.In parallelogram PQRS, angle P = 65°. Which reason correctly explains why angle R = 65°?

9.Point A and point B are 5 cm apart. How many points are exactly 4 cm from A and exactly 3 cm from B?

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

11.In triangle ABC, angle A is 2x°, angle B is 3x° and angle C is 4x°. Work out the size of angle B.

12.A cylinder has a volume of 942 cm³ and a height of 12 cm. Using π = 3.14, work out the radius of the cylinder.

13.A triangle has vertices A(1, 1), B(5, 1) and C(5, 4). Work out the lengths of AB, BC and CA, and write down whether triangle ABC is right-angled.

14.A right-angled triangle has its two shorter sides equal to 8 cm and 15 cm. Work out the value of cos θ, where θ is the angle opposite the 8 cm side. Give your answer as a fraction.

15.Which of these correctly compares a 'line' with a 'line segment'?