Geometry and measures worksheet — GCSE Higher

1.A student is proving that opposite angles of a cyclic quadrilateral ABCD sum to 180°, letting a = angle DAB and c = angle BCD, and using the fact that the angle at the centre is twice the angle at the circumference. She has already written: 'By the angle at the centre theorem, the reflex angle BOD equals 2a and the non-reflex angle BOD equals 2c.' Which statement must come immediately after this one in a correct proof?

2.What does the statement AB = CD mean, where A, B, C and D are all points?

3.A right-angled triangle has a hypotenuse of 14 cm. One of the other angles is 30°. Work out the exact length of the side opposite the 30° angle.

4.Triangle ABC has AB = 8 cm, BC = 10 cm and CA = 6 cm. Triangle LMN has LM = 10 cm, MN = 6 cm and NL = 8 cm. The two triangles are congruent by SSS. Which of the following correctly matches the corresponding vertices?

5.A hexagonal prism has 8 faces and 12 vertices. Using F + V − E = 2, work out how many edges it has.

6.A circle has circumference 31.4 cm. Using π = 3.14, work out the radius of the circle.

7.A footpath sign shows a distance of 0.45 km to the next village. Work out this distance in metres.

8.A triangle has vertices A(0, 0), B(10, 0) and C(5, 12). Work out the area of the triangle.

9.A circle has centre O. PQ, RS and TU are three chords of the circle, and only one of them passes through O. Write down what this tells you about that chord.

10.In triangle ABC, AB = 12 cm, AC = 7 cm and the area of the triangle is 33 cm². Given that angle BAC is obtuse, work out the size of angle BAC. Give your answer to 1 decimal place.

11.A right-angled triangle has two sides of length 5 cm and 12 cm, and the angle between those two sides is 90°. Work out the length of the hypotenuse.

12.An isosceles trapezium has exactly one pair of parallel sides, and its two non-parallel sides are equal in length. How many lines of symmetry does it have?

13.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″.

14.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.

15.Work out the length of the straight line segment joining the points (2, −4) and (−3, −4).