Geometry and measures worksheet — GCSE Higher

1.A zip-wire is fixed at an angle of 60° to the horizontal ground. The zip-wire is 12 m long. Using the exact value of cos 60°, work out the exact horizontal distance it covers.

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

3.A roof has two triangular sections, P and Q. Section P has a sloping edge of 3.6 m and a base edge of 2.4 m, with an angle of 70° between them. Section Q has a sloping edge of 3.6 m and a base edge of 2.4 m, with an angle of 70° between them, arranged the same way round. A roofer wants to cut identical triangular felt panels for both sections without measuring Section Q separately. Which condition confirms that the two sections are congruent?

4.Each of these has an exact value. Write down the one whose value is the greatest.

5.The bearing of a harbour B from a ferry's position A is 260°. What is the bearing of A from B?

6.From an external point P, two tangents PA and PB touch a circle with centre O at the points A and B. Angle APB = 46°. C is a point on the major arc AB, the arc on the opposite side of AB from P. Work out the size of angle ACB.

7.Triangle ABC is translated by the column vector with top number 3 and bottom number −5 to form triangle A′B′C′. Triangle A′B′C′ is then translated by the column vector with top number −7 and bottom number 2 to form triangle A″B″C″. Work out the single column vector that translates triangle ABC directly to triangle A″B″C″.

8.A tent's sloping side makes an angle of 45° with the ground. The sloping side is 3√2 m long. Using the exact value of sin 45°, work out the exact height of the tent.

9.A regular hexagon is divided into six identical triangles by joining its centre to each of the six vertices. Work out the size of the angle of one of these triangles at the centre of the hexagon.

10.Triangle ABC is isosceles, with AB = AC. Angle B = 70°. Which reason correctly explains why angle C = 70°?

11.Two straight paths cross at a junction in a park. One of the angles between them, measured with a protractor, is 76°. What is the size of the angle immediately next to it on a straight line?

13.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.

14.A solid has 6 faces: one pentagon and five triangles that all meet at a single point above the pentagon. What is the name of this solid?

15.Triangle ABC is right-angled at B. Triangle DEF is right-angled at E. AB = DE = 6 cm and AC = DF = 10 cm (AC and DF are the hypotenuses of their triangles). A student says this is not enough information to prove the triangles are congruent, because only two sides are given. Which reason shows the student is wrong?