Geometry and measures worksheet — GCSE Higher

1.Work out the exact value of sin 45° × cos 45°.

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

3.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.

4.Two sides of a triangle are 9 cm and 15 cm long. Write down which one of these lengths is possible for the third side.

5.A shape is translated by the vector (5, −3), and the image is then translated by the vector (−5, 3). Which single transformation has the same overall effect as this combination, for every point?

6.A solid has 5 faces: one square face and four triangular faces, which all meet at a single point above the square. What is the name of this solid?

7.A tangent to a circle touches the circle at exactly one point, P. Work out the size of the angle between the tangent and the radius drawn to P.

8.Shape S is enlarged by a scale factor of 3, centre the origin. A vertex of S is at (2, 1). Work out the coordinates of the image of this vertex.

9.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?

10.A straight line crosses a pair of parallel lines. One of the co-interior (allied) angles is 118°. Work out the size of the other co-interior angle.

11.In triangle XYZ, the side opposite vertex X is labelled using the standard lowercase-letter convention. Which label is correct?

12.Point P has coordinates (4, 2). P is rotated 90° clockwise about the point (1, 1), and the image is then reflected in the line x = 1. Work out the coordinates of the final image of P.

13.A picture frame is a rhombus with a diagonal of 16 cm and a diagonal of 12 cm. The two diagonals meet at right angles at their midpoints. Work out the length of one side of the rhombus.

14.Triangle T has a vertex A at (3, 1). It is enlarged by a scale factor of −2, centre (1, 1). Work out the coordinates of the image of point A.

15.Triangles LMN and XYZ have LM = XY, MN = YZ and LN = XZ. Give a reason why triangle LMN is congruent to triangle XYZ.