Geometry and measures worksheet — GCSE Higher

1.A radar station is at the origin of a grid, where each unit represents 1 km. A boat is detected at the point (7, 24). Work out the boat's distance from the station, then work out how many hours it will take the boat to reach the station travelling directly towards it at 5 km per hour.

2.In cyclic quadrilateral ABCD, with vertices in that order around the circle, the diagonal BD is drawn. In triangle ABD, angle ABD = 35° and angle ADB = 65°. Side DC is extended beyond C to a point E. Work out the size of angle BCE.

3.A cylinder has a radius of 5 cm. Its volume is 471 cm³. Using π = 3.14, work out the height of the cylinder.

4.m is the column vector with top number 4 and bottom number 6. n is the column vector with top number −6 and bottom number −9. Given that n = k × m for some number k, work out the value of k.

5.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?

6.Triangle T has vertices (1, 1), (3, 1) and (1, 4). It is mapped onto triangle T′ with vertices (5, −1), (3, −1) and (5, −4). Which single composition of two transformations maps T onto T′?

7.A, B and C are points on a circle with centre O, and AB is a diameter. Prove that angle ACB = 90°, using the fact that triangle OAC and triangle OBC are both isosceles. In the proof, angle OAC = angle OCA = x and angle OBC = angle OCB = y. Which equation correctly expresses the angle sum of triangle ABC in terms of x and y, and leads to the required proof?

8.A shape is reflected in the line y = 1, and the image is then reflected in the line y = 4. Which single transformation is equivalent to this combination, for every point?

9.a is the column vector with top number 3 and bottom number −2. b is the column vector with top number −1 and bottom number 5. Work out 2a + b, giving your answer as a column vector in the form (top, bottom).

10.From an external point P, two tangents PA and PB touch a circle with centre O at A and B. Angle APB = 40°. Work out the size of angle AOB.

11.Two sides of a triangle are 20 cm and 8 cm long. Write down which one of these lengths is possible for the third side.

12.A rectangle has vertices A(1, 1), B(4, 1), C(4, 5) and D(1, 5). Work out the perimeter of the rectangle.

13.Point P has coordinates (6, 4). P is rotated 90° clockwise about the point (1, 2), and the image is then translated by the vector (−3, 5). Work out the coordinates of the final image of P.

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

15.A student draws a net using 5 identical squares arranged in a row of four with one extra square attached to the side of one of them. Can this net be folded to make a closed cube?