Geometry and measures worksheet — GCSE Higher

1.Two similar triangular flower beds have lengths in the ratio 3 : 7. The area of the smaller bed is 45 m². Work out the area of the larger bed.

2.A shape is reflected in the line y = −x. Which of these points is invariant under this reflection?

3.The bearing of a campsite B from a walker's position A is 070°. What is the bearing of A from B?

4.A cuboid has length 3 cm, width 4 cm and height 12 cm. Work out the length of the diagonal that runs from one corner of the cuboid to the opposite corner.

5.In triangle OAB, OA = a and OB = b. P is the point on OA such that OP = (1/3)a, and Q is the point on OB such that OQ = (1/3)b. Express the vector PQ in terms of a and b.

6.A rotation of 200° clockwise about the origin is applied, and the image is then rotated 250° anticlockwise about the origin. Work out the single angle of rotation, measured clockwise and between 0° and 360°, that has the same overall effect as the two rotations combined.

7.A student is proving that opposite angles of a cyclic quadrilateral ABCD sum to 180°, letting a = angle DAB and c = angle BCD, and using the fact that the angle at the centre is twice the angle at the circumference. She has already written: 'By the angle at the centre theorem, the reflex angle BOD equals 2a and the non-reflex angle BOD equals 2c.' Which statement must come immediately after this one in a correct proof?

8.A, B, C and D are points on a circle with centre O, in that order around the circle. B lies on the major arc AC, and angle AOC = 152°. In triangle ACD, angle ACD = 30°. Work out the size of angle DAC.

9.Triangle ABC is similar to triangle PQR. Side AB is 8 cm and corresponds to side PQ, which is 12 cm. Side AC is 6 cm and corresponds to side PR. Work out the length of PR.

10.A, B and C are points on a circle with centre O. B is on the major arc AC. Angle AOC = 116°. Work out the size of angle ABC.

11.A postcard measuring 10 cm by 15 cm is enlarged by a scale factor of 2 to make a poster. Work out the area of the poster, in cm².

12.Triangle ABC has AB = 5 cm and BC = 7 cm. Triangle DEF has DE = 5 cm and EF = 7 cm. Which extra fact would prove the triangles are congruent by SAS?

13.A right-angled triangle has a hypotenuse of 29 cm and one shorter side of 20 cm. Work out the length of the other shorter side.

14.Lines GH and JK are parallel. A straight line crosses GH at point M and crosses JK at point N. Angle HMN = 71°. Angle HMN and angle MNK are alternate angles. Work out angle MNK.

15.A quadrilateral has both pairs of opposite sides parallel and equal in length, but none of its angles are right angles. Which statement about this quadrilateral is correct?