Geometry and measures worksheet — GCSE Higher

1.A scale model of a bridge is built so that 2 cm on the model represents 5 m of the real bridge. The real bridge is 45 m long. Work out the length of the model bridge, in centimetres.

2.P is the point (3, 4) and Q is the point (6, 0). Work out the distance of each point from the origin, and write down which point is closer to the origin.

3.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.

4.In a kite, one pair of opposite angles are equal in size. In kite WXYZ, angle X = 40° and angle Z = 100° are the two angles that are NOT equal to each other. The other two angles, W and Y, are equal to each other. Work out the size of angle W.

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

6.A cuboid has three different edge lengths: a length, a width and a height, all different from each other. How many rectangles make up its net in total?

7.Triangle J is mapped onto triangle K by a single transformation with centre O. Triangle K is half the size of triangle J and is inverted (turned through 180°). Which of these transformations maps J onto K?

8.Shape T has a vertex at (5, 3). T is enlarged by scale factor −1, centre (2, 1). Work out the coordinates of the image of the vertex (5, 3).

9.A square-based pyramid has a square base and four identical triangular sloping faces, with its apex directly above the centre of the base. How many planes of symmetry does it have?

10.A vertical flagpole casts a shadow 6 m long at the same time as a 1.2 m vertical post casts a shadow 1.8 m long. The flagpole and the post are both vertical, and the triangles formed by each object, its shadow, and the sun's ray are similar. Work out the height of the flagpole.

11.A ferris wheel's circular frame has centre O. A support strut runs from O to a point A on the rim, and a second strut runs from O to point B on the rim, with angle AOB = 76°. A cabin is mounted at point C on the major arc AB, and D is a separate point on the minor arc AB. Work out the difference between angle ACB and angle ADB.

12.A skateboard ramp is designed so that a straight rail RT is tangent to a circular curve at the point T, and TC is a chord of the curve from T to another point C on the curve. A second rail continues straight through T on the other side, so that the two rails together form a straight line, and the angle between that second rail and the chord TC is 109°. A support point D is placed on the curve, on the major arc TC. Work out the size of angle TDC, the angle subtended by the chord TC at D.

13.A tank holds 3.2 m³ of water. Work out this volume in litres.

14.A shape is reflected in the x-axis and the image is then reflected in the y-axis. Which single transformation is equivalent to this combination for every point?

15.A(1, 2), B(5, 2) and C(5, 6) are three of the four vertices of a square ABCD. Work out the coordinates of D.