Geometry and measures worksheet — GCSE Higher

1.In pentagon PQRST, which of the following correctly names the interior angle at vertex R, using standard three-letter angle notation?

2.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?

3.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?

4.In triangle ABC the angle at A and the angle at C are equal. The side AB is extended beyond B, and the exterior angle formed at B measures 98°. Work out the size of the angle at A.

5.State whether the line segment joining A(−4, 3) and B(2, 6) is horizontal, vertical or neither, and give a reason for your answer.

6.A triangular flag has a base of 40 cm and a perpendicular height of 25 cm. Work out its area.

7.A hiker starts by walking on a bearing of 245°. She turns clockwise through 50°, then turns clockwise through a further 15°, and continues walking in a straight line. What bearing is she now walking on?

8.A, B, C and D are points on a circle with centre O, placed in that order around the circle so that ABCD is a cyclic quadrilateral. B lies on the major arc AC and D lies on the minor arc AC. Angle AOC = 104°. Work out the size of angle ADC.

9.Shape S has an area of 5 cm². Shape S is enlarged by a scale factor of −3 to give shape T. Work out the area of shape T.

10.A gardener wants to plant a tree so that it is the same distance from two straight garden walls that meet at a corner, and also the same distance from two ornamental posts standing 4 m apart. Which construction locates this point?

11.A circle has a radius of 1 cm. Work out the circumference of the circle. Give your answer in terms of π.

12.Triangle ABC has AB = 6 cm and BC = 9 cm, with angle A = 40°. Triangle DEF has DE = 6 cm and EF = 9 cm, with angle D = 40°. Meera says these facts prove the triangles are congruent by SAS. Is Meera correct?

13.OABC is a parallelogram, with OA = a and OC = c. M is the midpoint of OC, and N is the point on AC such that AN is twice NC. By finding the vectors MN and MB, show that M, N and B are collinear, and give the scalar k such that MN = k × MB.

14.Triangle DEF is isosceles, with DE = DF. Angle E = 58°. Work out angle F.

15.A chord divides a circle into two segments of different sizes. Write down the name given to the smaller of the two segments.