Geometry and measures worksheet — GCSE Higher

1.Put sin 30°, tan 30° and cos 30° in order of size, starting with the smallest.

2.A model of a bridge is built to a scale where every length on the model is 1/240 of the matching length on the real bridge. A support beam on the model measures 3.6 cm. Work out the length of the real support beam, giving your answer in metres.

3.A tangent to a circle with centre O touches the circle at point P, where OP = 5 cm. Point Q lies on the tangent so that PQ = 12 cm. Using the fact that a tangent is perpendicular to the radius at the point of contact, work out the length OQ.

4.A hiker walks on a bearing of 065°. She then turns clockwise through 90° and continues walking in a straight line. What bearing is she now walking on?

5.A transversal crosses a pair of parallel lines. At one line, the angle is (5x + 4)°. The corresponding angle at the other line is (3x + 24)°. Work out the value of x.

6.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.

7.A quadrilateral has both pairs of opposite sides parallel and equal in length, but none of its angles are right angles. Which statement about this quadrilateral is correct?

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

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

10.A shape is reflected in the x-axis and the image is then reflected in the y-axis. Which single transformation is equivalent to this combination for every point?

11.The point A(3, 2) is translated by the vector (−5, 4) to give point B. Work out the coordinates of B.

12.Work out the gradient of the line segment joining A(2, 3) and B(6, 11).

13.Two hikers walk from a campsite A to a lookout B, a distance of 14 km on a bearing of 038°. From B they walk to a shelter C, a distance of 20 km on a bearing of 142°. Work out the direct distance from A to C. Give your answer to 1 decimal place.

14.A mast stands vertically on level ground and its top is 60 m above the ground. Amelia and Noah stand on the ground on the same side of the mast, in line with its foot. The angle of elevation of the top of the mast is 30° from where Amelia stands and 60° from where Noah stands. Work out the distance between Amelia and Noah. Give your answer to 1 decimal place.

15.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.