Robinson Tuition
Grade 5 · Foundation and Higher · free GCSE maths worksheet

Loci

Regions defined by distance rules.

10 original questions, 25 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Sam is asked to shade the region of all points that are less than 4 cm from a point P.

Sam shades a square with sides of length 8 cm, with P at the centre of the square.

Explain why Sam's region is wrong.

1 mark
Show answer
The set of points less than 4 cm from P is the inside of a circle, centre P, radius 4 cm - not a square. The corners of Sam's square are further than 4 cm from P.
2

Describe fully the locus of all points that are exactly 5 cm from a fixed point A.

1 mark
Show answer
A circle with centre A and radius 5 cm
3

Points A and B are 8 cm apart.

Describe how to construct the perpendicular bisector of the line segment AB, using only a pair of compasses and a straight edge.

3 marks
Show answer
Compasses open to more than half of AB. Draw arcs above and below AB from centre A, then with the same radius from centre B, so they cross at two points. Join the two crossing points with a straight line.
4

Two towns, Ashford and Bexley, are 60 km apart.

A new phone mast must be built so that it is exactly the same distance from Ashford as it is from Bexley.

Describe fully the locus of all possible positions of the mast.

2 marks
Show answer
The perpendicular bisector of the line segment joining Ashford and Bexley - the straight line at right angles to it, passing through its midpoint, 30 km from each town
5

On a coordinate grid, A is the point (2, 1) and B is the point (2, 9).

The region R contains all the points that are closer to A than to B.

  1. Write down the equation of the boundary of R.
  2. Is the point (7, 6) in R? Give a reason for your answer.
2 marks
Show answer
a) y = 5 b) No, because it is above the line y = 5, so it is closer to B
6

Two straight paths, PQ and PR, meet at the point P. The angle QPR is 70°.

A bench is placed so that it is the same distance from both paths.

  1. What is the name of the line on which the bench must lie?
  2. Work out the size of the angle between this line and the path PQ.
2 marks
Show answer
a) The angle bisector of QPR b) 35 degrees
7

A goat is tethered to a post in the middle of a large flat field by a rope of length 7 m.

The rope stays taut or slack, so the goat can reach any point up to 7 m from the post.

  1. Describe fully the boundary of the region the goat can reach.
  2. Work out the area of grass the goat can graze. Give your answer correct to 1 decimal place.
3 marks
Show answer
a) a circle, centre the post, radius 7 m b) 153.9 m²
8

A straight path AB is 10 m long. A sprinkler system waters every point of the ground that is within 2 m of the path, including both ends.

  1. Describe the shape of the watered region.
  2. Work out the area of the watered region. Give your answer correct to 3 significant figures.
4 marks
Show answer
a) A rectangle 10 m by 4 m with a semicircle of radius 2 m added at each end b) 52.6 m2
9

A rectangular shed measures 10 m by 8 m, and stands in a large flat field. A goat is tied to one corner of the shed, on the outside, with a rope 6 m long. The rope is too short to reach round either end of the shed.

Work out the area of grass the goat can reach. Give your answer correct to 3 significant figures.

3 marks
Show answer
84.8 m2
10

The diagram shows a scale drawing of a rectangular garden ABCD, with AB = 10 cm and BC = 6 cm on the drawing.

The scale is 1 cm to 2 m.

A pond will be dug in the garden so that it is

more than 6 m from corner A

and closer to side AB than to side DC.

Shade the region of the garden where the pond can be dug. You must show all your construction lines.

ABCD10 cm6 cm
4 marks
Show answer
Shade the part of the garden between AB and the mid-line parallel to AB (the line 3 cm from AB on the drawing) and outside the arc of radius 3 cm centred at A (6 m at scale 1 cm : 2 m)

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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