Geometry and measures worksheet — GCSE Higher

1.A rectangular garden gate is 3 m wide and 4 m high. A straight brace runs from one corner of the gate to the opposite corner. Work out the length of the brace.

2.Triangle ABC and triangle DEF are similar. The area of triangle ABC is 36 cm² and the area of triangle DEF is 100 cm². Work out the ratio of the lengths of triangle ABC to the lengths of triangle DEF, in its simplest form.

3.Two hikers walk from a campsite A to a lookout B, a distance of 14 km on a bearing of 038°. From B they walk to a shelter C, a distance of 20 km on a bearing of 142°. Work out the direct distance from A to C. Give your answer to 1 decimal place.

4.A written description of a diagram states: 'AB is marked with a single arrow, CD is marked with a single arrow, and EF is marked with a double arrow.' Using the standard convention for marking parallel lines, what can be concluded?

5.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.

6.A cuboid-shaped shipping crate has a length of 12 m, a width of 5 m, and a volume of 360 m³. Work out the height of the crate, in metres.

7.A tangent touches a circle with centre O at the point P. Q is a point on the tangent. Write down the circle fact that tells you the size of angle OPQ.

8.In triangle ABC, AB = x cm, AC = (x + 2) cm, BC = 15 cm and angle BAC = 100°. Work out the value of x. Give your answer to 1 decimal place.

9.A photograph is 10 cm wide and 15 cm tall. It is enlarged to make a similar poster that is 40 cm wide. The poster costs £0.80 per centimetre of its height to print, based on its full height. Work out the cost of printing the poster.

10.Shape S has an area of 48 cm². It is enlarged by a scale factor of 1/4 to give shape T. Work out the area of shape T.

11.Work out the exact value of (sin 30°)² + (cos 30°)².

12.In triangle ABC, AB = 8 cm, AC = 7 cm and the area of the triangle is 24 cm². Given that angle BAC is acute, work out the size of angle BAC. Give your answer to 1 decimal place.

13.In parallelogram PQRS, angle P = 65°. Which reason correctly explains why angle R = 65°?

14.The sum of the interior angles of a polygon is 1980°. Work out the number of sides of the polygon.

15.From an external point P, two tangents PA and PB touch a circle with centre O at points A and B. Angle APB = 44°. C is a point on the major arc AB. Using the fact that PA = PB, and the alternate segment theorem, work out the size of angle ACB.