Geometry and measures worksheet — GCSE Higher

1.OABC is a parallelogram, with OA = a and OC = c. M is the midpoint of OC, and N is the point on AC such that AN is twice NC. By finding the vectors MN and MB, show that M, N and B are collinear, and give the scalar k such that MN = k × MB.

2.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?

3.A solid is built from centimetre cubes: a base layer of 2 rows of 3 cubes each (a 3 by 2 rectangle of cubes), with one extra cube placed on top of one corner cube of that base. Looking down from directly above (the plan view), how many squares are visible?

4.Shape S is enlarged by a scale factor of −3 about a fixed centre. Which statement correctly describes the image compared to the original shape?

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

6.A garden flower bed is in the shape of a trapezium. Its two parallel edges are 3.5 m and 6.5 m long, and the perpendicular distance between them is 4 m. Bedding plants cost £2.50 per square metre to cover the bed. Work out the total cost of the bedding plants.

7.Triangle ABC has AB = 6 cm, BC = 8 cm and angle B = 90°. Triangle XYZ has XY = 6 cm, YZ = 8 cm and angle Y = 90°. Which congruence statement correctly shows the matching vertices?

8.A locus consists of all the points that are the same perpendicular distance from two parallel lines 6 cm apart. Describe this locus.

9.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?

10.Point Q has coordinates (3, 5). It is translated by the vector (1, −2), and the image is then enlarged by scale factor 3, centre the origin. Work out the coordinates of the final image of Q.

11.A sailmaker cuts two triangular sail panels from a pattern. Panel X has a 10 m side, with angles of 50° and 75° at its two ends. Panel Y also has a 10 m side, with angles of 50° and 75° at its two ends, in the same arrangement. The sailmaker wants to check the panels will match exactly, without cutting a third measurement. Which condition proves the two panels are congruent?

12.A regular polygon has an exterior angle of 45°. Work out the number of sides of the polygon.

13.In triangle ABC, AB = x cm, AC = (x + 3) cm, BC = 7 cm and angle BAC = 60°. Work out the value of x.

14.In triangle OAB, OA = a and OB = b. P is the point on OA such that OP = (1/3)a, and Q is the point on OB such that OQ = (1/3)b. Express the vector PQ in terms of a and b.

15.Two spheres are similar solids. Their radii are in the ratio 3 : 4. Write down the ratio of their surface areas.