Geometry and measures worksheet — GCSE Higher

1.A tile is described as having two pairs of parallel sides, four equal sides, and diagonals that cross at right angles but are not equal in length. Give a reason why this tile cannot be a square.

2.An enlargement has a scale factor of −2. Priti says the image must be a mirror image of the original shape. Is Priti correct?

3.From an external point P, two tangents PA and PB touch a circle with centre O at the points A and B. Angle APB = 50°. Work out the size of angle OAB.

4.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.

5.Two similar triangles have lengths in the ratio 2 : 5. The sides of the smaller triangle are 4 cm, 6 cm and 8 cm. Work out the length of the longest side of the larger triangle.

6.The diagram shows the plan view and the front elevation of a cuboid-shaped box, each drawn as a rectangle. Work out the area of the box's side elevation, in square centimetres.

7.A solid cube has a volume of 8 cm³. Work out the length of one edge of the cube.

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.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.

10.A designer plots a logo point at (3, 4) on a grid. She reflects it in the line y = 1 and then rotates the image 90° clockwise about the point (1, 1) to create a repeating tile pattern. Work out the coordinates of the point after both transformations.

11.A water tank is a cube with edges of length 2 m. Work out how many cubic centimetres the tank holds when it is full.

12.A straight line crosses two parallel lines. One of the angles formed is (2x + 10)°, and the angle alternate to it is 74°. Work out the value of x.

13.An enlargement with scale factor 5 maps the point (4, 1) to the point (12, 9). Which of these points is invariant (does not move) under this enlargement?

14.Vertex X of a triangle is at (−3, 5). After a translation, the image of X is at (2, −1). Write down the column vector of this translation.

15.A cube has edges of length 5 cm. Work out the length of the diagonal that runs from one corner of the cube to the opposite corner. Give your answer as a surd in its simplest form.