15.7 Loci and Regions

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: drawing loci from points and lines, and shading regions that satisfy given rules. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Draw the locus of points a fixed distance from a point or from a line segment
• Draw the locus of points equidistant from two points or from two lines
• Shade a region that satisfies one or more conditions
• Solve real-life problems with loci, including scale drawings

Key words

• Locus (loci): The set of all points that follow a rule. Often a line or a curve.
• Region: An area containing all the points that follow a rule.
• Equidistant: The same distance from two or more things.
• Boundary: The edge of a region. Dashed if points on it are not included.

Fixed distance from a point or a line

Points that are exactly 2.5 cm from a point P lie on a circle of radius 2.5 cm. Points less than 2.5 cm from P are inside the circle.

Points a fixed distance from a line segment AB form a "running track" shape: two straight lines parallel to AB, joined by semicircles centred on A and B.

Worked example: a goat on a rope

A goat is tied to a post with a 6 m rope in the middle of a large field. Describe the region it can reach, and its area.
Every point it reaches is 6 m or less from the post: a circle of radius 6 m and its inside.
Area = π × 6² = 36π ≈ 113.1 m²

Answer: a disc of radius 6 m, area about 113 m²

Equidistant from two points or two lines

Points the same distance from A and B lie on the perpendicular bisector of AB. Points the same distance from two lines that meet lie on the angle bisector.

Worked example: closer to one point

Shade the points that are closer to B than to A.
Construct the perpendicular bisector of AB. It splits the page into two halves.

Answer: shade the side of the bisector that contains B

Regions with more than one rule

Draw the locus for each rule, then decide which side of each boundary you need. Shade the part that obeys all the rules and label it.

Worked example: a sprinkler in a garden

ABCD is a rectangular garden 8 m by 5 m. A sprinkler at A waters everything up to 4 m away. Shade the watered part of the garden that is closer to AB than to AD.
Rule 1: within 4 m of A, so inside a quarter circle of radius 4 cm on the drawing.
Rule 2: closer to AB than AD, so below the bisector of angle A (45°).

Answer: region R: the 45° slice of the circle between AB and the bisector

Key fact: Use a solid line when points on the boundary count ("at most 4 m") and a dashed line when they do not ("less than 4 m", "closer to").

Common mistakes

• Drawing a rectangle around a line segment. The ends must be semicircles.
• Using a straight line between two points instead of the perpendicular bisector.
• Forgetting to use the scale: 3 m at 1 cm to 2 m is a radius of 1.5 cm.
• Shading the region that breaks a rule instead of the one that obeys it. Test a point in your shaded region.

Revision questions

Try these, then check your answers below each one.

Question 1. What is the locus of points that are exactly 3 cm from a point P? (Choose from: A circle of radius 3 cm centred on P, A square of side 3 cm, A straight line 3 cm from P, A circle of radius 6 cm centred on P.)

Answer: A circle of radius 3 cm centred on P

Working: All points 3 cm from P make a circle of radius 3 cm centred on P.

Question 2. What is the locus of points that are the same distance from two lines that meet at a point? (Choose from: The bisector of the angle between the lines, The perpendicular bisector of the lines, A circle, A line parallel to one of the lines.)

Answer: The bisector of the angle between the lines

Working: Points equidistant from two meeting lines lie on the angle bisector.

Question 3. A goat is tied to a post in the middle of a field with a rope 4 m long. Work out the area of grass it can reach, to 1 decimal place.

Answer: 50.3 m²

Working: π × 4² = 16π = 50.3 m² (1 d.p.).

Related lessons

• Previous lesson: 15.6 Constructions
• Next lesson: 15.8 Bearings
• In this chapter: 15.1 3D Solids
• In this chapter: 15.2 Plans and Elevations
• In this chapter: 15.3 Accurate Drawings 1

Continue in the full course

This lesson is section 15.7 (EGCMF1507) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.