Geometry and measures worksheet — GCSE Higher

1.A path goes from A(1, 1) to B(1, 5), then from B to C(6, 5). Work out the total length of the path from A to C.

2.AB is a diameter of a circle with centre O, and C is a point on the circle. A student writes four statements to prove that angle ACB = 90°. Statement 1: OA = OC, since both are radii, so triangle OAC is isosceles with angle OAC = angle OCA. Statement 2: OB = OC, since both are radii, so triangle OBC is isosceles with angle OBC = angle OCB. Statement 3: in triangle ABC, the three angles sum to 360°, so angle OAC + angle OBC + angle ACB = 360°, meaning 2 × angle ACB = 360° and angle ACB = 180°. Statement 4: A, O and B lie on a straight line, since AB is a diameter through the centre O. Which one of these four statements is mathematically incorrect?

3.In triangle OAB, OA = a and OB = b. P lies on AB such that AP is twice PB. Express the vector OP in terms of a and b.

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

5.The interior angles of a pentagon are 100°, 110°, 120°, x° and x°. Work out the size of each of the two angles marked x°.

6.An enlargement with scale factor 5 maps the point (4, 1) to the point (12, 9). Which of these points is invariant (does not move) under this enlargement?

7.O is the centre of a circle, and PT is a tangent to the circle at the point T. A is a point on the circle, with OA and OT both radii and angle AOT = 122°. At the point T, the chord TA lies between the radius TO and the tangent TP. Work out the size of angle ATP, the angle between the chord and the tangent.

8.A cable supporting a flagpole is anchored to the ground 5 m from the base of the pole. The cable makes an angle of 60° with the ground. Using the exact value of tan 60°, work out the exact height of the flagpole.

9.ABCD is an isosceles trapezium. The sides AB and DC are parallel, and the two sloping sides AD and BC are equal in length. Angle D is 112°. Work out the size of angle B.

10.A roof truss has two horizontal parallel rafters, one above the other. A straight strut crosses both rafters. Where the strut crosses the lower rafter, the angle above the rafter and to the left of the strut is 65°. Work out the size of the angle above the upper rafter and to the right of the strut, where the strut crosses it.

11.Work out the exact value of sin 45° × cos 45°.

12.A circular plate has a diameter of 20 cm. Work out the area of the plate. Use π = 3.14.

13.A dog is tied by a lead 4 m long to a fixed ring on a straight garden wall. The wall extends much further than the lead can reach in both directions, and the dog cannot cross through it. Describe the region the dog can reach.

14.A student is asked to rearrange the cosine rule a² = b² + c² − 2bc cos A to make cos A the subject. They write: cos A = (a² − b² − c²) / (2bc). Is the student's rearrangement correct?

15.Triangle T has a vertex at (2, 1). It is enlarged by a scale factor of 3, centre (0, 0). Work out the coordinates of the image of this vertex.