G2Geometry & MeasuresFoundation & HigherGrades 3–6

Constructions & Loci

A phone mast must be within 3 km of a village but closer to road A than road B. Where can it go? Questions like this are answered with a ruler and a pair of compasses. A $\textbf{locus}$ is the set of all points that obey a rule, and the constructions in this lesson draw loci accurately.

45 min AQA GCSE Maths

The big picture

Four constructions do almost all of the work: the perpendicular bisector of a line, the bisector of an angle, the perpendicular from a point to a line, and the perpendicular at a point on a line. Each one is built from arcs of equal radius.

Each standard locus matches a construction. Equidistant from two points is a perpendicular bisector; equidistant from two lines is an angle bisector. Combining loci and shading the region that satisfies every condition is the typical exam question.

What you'll learn

Construct perpendicular bisectors and angle bisectors
Construct perpendiculars from and at a point
Draw loci of points a fixed distance from a point or a line
Find and shade regions that satisfy several conditions

Perpendicular bisector

Set the compasses to more than half the length of $AB$ .

Draw arcs above and below the line from $A$ , then from $B$ with the same radius.

Join the two crossing points. This line cuts $AB$ in half at $90^{\circ}$ .

Your turnSame method, new numbers — try it before moving on.

What is the name of the line containing every point that is the same distance from two fixed points $A$ and $B$ ?

Your answer

Show worked solution

Solution

1Points equidistant from $A$ and $B$ lie on the line that cuts $AB$ in half at right angles.
2That line is the perpendicular bisector.

Answer: The perpendicular bisector of $AB$

Why this works

Every point on the perpendicular bisector is the same distance from $A$ as from $B$ . That is why it is also the locus of points equidistant from two points.

Angle bisector and perpendiculars

Angle bisector: from the vertex, draw an arc cutting both arms. From each crossing point, draw arcs of equal radius that meet. Join the vertex to that point.

Perpendicular from a point $P$ to a line: draw an arc from $P$ cutting the line twice, then construct the perpendicular bisector of those two points.

An angle of $60^{\circ}$ comes from an equilateral triangle (same radius throughout); bisecting it gives $30^{\circ}$ .

Tip — Leave all construction arcs visible. Rubbing them out removes the evidence the marks depend on.

The standard loci

A fixed distance from a point: a circle.

A fixed distance from a line segment: two parallel lines joined by semicircles at the ends.

Equidistant from two points: the perpendicular bisector.

Equidistant from two intersecting lines: the angle bisector.

Example 1A rectangular garden

ABCD

is drawn to scale, 1 cm to 2 m. A tree is planted less than 6 m from

A

and closer to

AB

than to

AD

. Describe how to find the region.

16 m is 3 cm: draw an arc of radius 3 cm centred at

A

2Construct the bisector of angle

DAB

3The region is inside the arc on the

AB

side of the bisector.

AnswerShade the part of the quarter-circle of radius 3 cm that lies between the bisector and

AB

Your turnSame method, new numbers — try it before moving on.

A goat is tied to a post in the middle of a large field by a rope 5 m long. What area can it graze, to 1 decimal place?

Your answer

Show worked solution

Solution

1The locus of points within 5 m of the post is a circle of radius 5 m.
2Area $= \pi \times 5^2 = 78.54\ldots$ , so $78.5$ m².

Answer: $78.5$ m²

Tip — Check whether a boundary is included: "less than" gives a dashed boundary, "at most" a solid one.

Think like an examiner

Use compasses — freehand arcs lose marks.
Leave construction arcs visible.
Convert distances with the scale before drawing.

Remember these

\text{distance } r \text{ from a point} \to \text{circle}

Locus from a point.

PA = PB \to \text{perpendicular bisector}

Equidistant from two points.

\text{equidistant from two lines} \to \text{angle bisector}

Equidistant from two lines.

Watch out for these

Changing the compass width halfway through a bisector.

Keep the same radius for each pair of arcs.

Drawing a locus from a line as a rectangle.

The ends are semicircles.

Measuring angles with a protractor in a construction question.

Use only ruler and compasses.

Shading the wrong side of a boundary.

Test a point in the region against every condition.

Stretch yourself

A goat is tied to one corner of a rectangular shed measuring 4 m by 3 m, with a rope 5 m long. The goat stays outside the shed. Find the area it can graze, in terms of $\pi$ and to 3 significant figures.

Hint — Sketch the locus: a large circular sector, plus smaller sectors where the rope bends around the shed’s corners.

Reveal solution

SolutionAway from the shed the goat sweeps three-quarters of a circle of radius 5:

\tfrac34 \times 25\pi = 18.75\pi

. Around the 4 m side, 1 m of rope remains: a quarter circle of area

\tfrac14 \pi

. Around the 3 m side, 2 m remains: a quarter circle of area

\tfrac14 \times 4\pi = \pi

. Total

= 20\pi \approx 62.8

m².

\blacksquare

Questions students ask

What is the difference between a locus and a region?

A locus is a line or curve of points; a region is an area of points.

How wide should I set my compasses?

For bisectors, more than half the length of the line; the exact width does not matter as long as it stays fixed.

Do I need to describe constructions in words?

Usually no — the drawing with visible arcs is the answer.

Key takeaways

Perpendicular bisector: equidistant from two points.
Angle bisector: equidistant from two lines.
Fixed distance from a point: a circle; from a line: a stadium shape.
Keep arcs visible and use scales carefully.
Regions combine several loci.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 32 marks

Describe the locus of points that are exactly 3 cm from a fixed point $P$ .[2]
Show mark scheme
- B1A circle
- B1Centre $P$ , radius 3 cm

Question 2Grade 53 marks

A rectangular field $ABCD$ is 40 m by 25 m, with $AB = 40$ m. A sprinkler at the midpoint of $AB$ waters every point of the field within 12 m of it.

Work out the area of the field that is watered, to the nearest square metre.[3]
Show mark scheme
- M1Recognises the watered region is a semicircle of radius 12 m inside the field
- M1 $\frac{1}{2} \times \pi \times 12^2$
- A1 $226$ m²
Answer: $226$ m²

Question 3Grade 63 marks

Triangle $ABC$ is drawn accurately.

Explain how you would find the region of points inside the triangle that are closer to $AB$ than to $AC$ , and less than 4 cm from $A$ .[3]
Show mark scheme
- B1Construct the bisector of angle $BAC$
- B1Draw an arc of radius 4 cm with centre $A$
- B1Shade the region on the $AB$ side of the bisector and inside the arc

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 14 lessons