Geometry and measures worksheet — GCSE Higher

1.A patio plan states: 'Side AB of the patio is parallel to side DC, and AB is longer than DC. Sides BC and AD are not parallel to each other and are not equal in length.' The four sides measure AB = 5.1 m, BC = 3.2 m, DC = 2.8 m and AD = 4.5 m. The tiler needs to order edging strip only for the sides that are not parallel to any other side. What total length of edging strip, in metres, should the tiler order?

2.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.

3.A gardener wants to plant a tree so that it is the same distance from two straight garden walls that meet at a corner, and also the same distance from two ornamental posts standing 4 m apart. Which construction locates this point?

4.Point A(1, 2) is enlarged to give image point A′(7, −4). The centre of enlargement lies on the x-axis. Work out the scale factor of the enlargement.

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

6.The perimeter of a square is 20 cm. Work out the area of the square.

7.An ice cream is made from a cone of radius 3 cm and height 10 cm, topped with a hemisphere of the same radius sitting exactly on top of the cone. Work out the total volume of the ice cream. Use π = 3.14. Give your answer to the nearest whole number. Volume of a cone = 1/3 × πr²h. Volume of a sphere = 4/3 × πr³.

9.Point D is at (3, 7). It is reflected in the line y = x. Work out the coordinates of the image of point D.

10.A conservatory has a roof panel shaped like a regular octagon. Work out the interior angle of the octagon, then use it to work out the size of the reflex angle at the same vertex, on the outside of the roof panel.

11.Two similar triangles have lengths in the ratio 3 : 4. The perimeter of the smaller triangle is 24 cm. Work out the perimeter of the larger triangle.

12.O is the centre of a circle, and PT is a tangent to the circle at the point T. A is a point on the circle, with OA and OT both radii and angle AOT = 122°. At the point T, the chord TA lies between the radius TO and the tangent TP. Work out the size of angle ATP, the angle between the chord and the tangent.

13.A flagpole is 12 m tall. Freya stands 9 m from the base of the flagpole on level ground. Work out the angle of elevation of the top of the flagpole from where Freya stands. Give your answer correct to 1 decimal place.

14.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?

15.In triangle OAB, OA = a and OB = b. M is the midpoint of OA, and N is the midpoint of OB. Express the vector MN in terms of a and b.