J560 · 8.02–8.05Angles & Geometric ReasoningFoundation & HigherGrades 3–6

Constructions & Loci

Q: How accurate do constructions need to be?

Typically within 2 mm for lengths and 2^ for angles.

With only a ruler and a pair of compasses you can bisect an angle, draw a perpendicular exactly through the middle of a line, and mark out every point a given distance from a landmark. These $\textbf{constructions}$ solve $\textbf{loci}$ problems: where can a goat tethered to a post graze? Which part of a park is closer to the café than the gate?

45 min Angles & Geometric Reasoning

The big picture

Constructions work because compasses draw points at exactly equal distances. Every standard construction uses that fact — two arcs of the same radius crossing at points equidistant from two things — which is why the arcs must be left visible as evidence.

A locus is a set of points obeying a rule. Four loci cover almost every GCSE question: a fixed distance from a point (circle), a fixed distance from a line (parallel lines with semicircular ends), equidistant from two points (perpendicular bisector), and equidistant from two lines (angle bisector).

Constructing a 30 degree angle - CorbettmathsCurated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

Before you start

Try a short calculation from the preceding topic without the notes.
Check the tier and grade band shown in this lesson.

Watch, pause, try

Video by corbettmaths. The video’s own board label is retained; follow the specification reference and scope notes on this lesson for assessed content.

Viewing focus: Construct the perpendicular bisector of a line segment

After watching, attempt these questions before opening the answers. You can also use the written lesson below.

After watching 1: The locus of points 3 cm from a point is:

Show answer and reasoning

A circle of radius 3 cm. All points at a fixed distance from a centre.

After watching 2: Points equidistant from $A$ and $B$ lie on:

Show answer and reasoning

The perpendicular bisector of $AB$ . By construction.

What you'll learn

Construct the perpendicular bisector of a line segment
Construct the bisector of an angle and a perpendicular from a point to a line
Construct triangles from given information
Solve loci problems, shading regions that satisfy several conditions

Perpendicular bisector

To bisect line segment $AB$ at right angles: set the compasses to more than half the length of $AB$ . Draw arcs from $A$ above and below the line, then, with the same radius, arcs from $B$ crossing them.

Join the two crossing points with a straight line. It passes through the midpoint of $AB$ at $90^{\circ}$ .

Every point on this line is the same distance from $A$ as from $B$ .

Tip — Keep the compass width fixed for both sets of arcs, and never rub out construction arcs — they are the evidence for the marks.

Angle bisector and perpendiculars

To bisect an angle at vertex $V$ : draw an arc centred at $V$ crossing both arms. From each crossing point, draw arcs of equal radius that meet inside the angle. Join $V$ to the meeting point.

Every point on the angle bisector is equidistant from the two arms.

To draw a 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; it passes through $P$ .

Why this works

The shortest distance from a point to a line is along the perpendicular. That is why "distance from a line" in loci problems always means the perpendicular distance.

Constructing triangles

Given three sides (SSS): draw one side accurately, then use compasses set to each of the other lengths from its ends; the arcs cross at the third vertex.

Given two sides and the included angle (SAS), or two angles and a side (ASA), use a protractor for the angles and a ruler for the sides.

Label vertices and leave all arcs visible.

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

Can a triangle be constructed with sides of 3 cm, 4 cm and 8 cm? Type yes or no.

Your answer

Loci and regions

Points a fixed distance from a point form a $\textbf{circle}$ . Points a fixed distance from a line segment form a "running track" shape: two parallel lines joined by semicircles.

Points equidistant from two points lie on the $\textbf{perpendicular bisector}$ . Points equidistant from two intersecting lines lie on the $\textbf{angle bisector}$ .

"Closer to $A$ than $B$ " means one side of the perpendicular bisector; "less than 3 m from $P$ " means inside a circle. Combine conditions by finding where all regions overlap, and shade that region.

Example 1A garden is a rectangle

ABCD

with

AB = 10

m and

BC = 6

m, drawn to a scale of 1 cm to 1 m. A tree is to be planted closer to

AB

than to

AD

and less than 4 m from

A

. Describe how to find the region.

1"Closer to

AB

than to

AD

": construct the bisector of angle

DAB

; the region is on the

AB

side.

2"Less than 4 m from

A

": draw an arc of radius 4 cm centred on

A

; the region is inside it.

3Shade where both conditions hold, inside the rectangle.

AnswerThe region inside the arc of radius 4 cm about

A

, between

AB

and the angle bisector of

DAB

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

A straight fence is 10 m long. Find the area of the region within 3 m of the fence (on both sides, including the rounded ends), to 1 decimal place.

Your answer

Tip — Convert real distances with the scale before drawing. With 1 cm to 2 m, a 5 m radius becomes 2.5 cm.

Think like an examiner

Leave all construction arcs visible.
Use a sharp pencil and draw lines accurately to within 2 mm and angles within $2^{\circ}$ .
Label and shade the final region clearly, with a key if needed.

Loci

\text{fixed distance from a point: circle}

Point locus.

\text{equidistant from two points: perpendicular bisector}

Two points.

\text{equidistant from two lines: angle bisector}

Two lines.

Watch out for these

Changing compass width between the two arcs of a bisector.

Use the same radius for both sets of arcs.

Measuring the midpoint with a ruler instead of constructing.

A construction question requires the arcs.

Drawing a fixed distance from a line segment without rounded ends.

The ends are semicircles centred on the endpoints.

Forgetting to apply the scale.

Convert real distances to map distances first.

Stretch yourself

Two phone masts $P$ and $Q$ are 8 km apart. A signal is received anywhere within 5 km of either mast. Explain which points on the perpendicular bisector of $PQ$ receive a signal, and find the length of that section.

Hint — Points on the perpendicular bisector are equidistant from $P$ and $Q$ . Use Pythagoras with half of $PQ$ .

Questions students ask

What equipment is allowed?

A ruler, compasses and pencil for constructions; a protractor for angle-based triangles and bearings.

What is the plural of locus?

Loci.

How accurate do constructions need to be?

Typically within 2 mm for lengths and

2^{\circ}

for angles.

Key takeaways

Constructions rely on equal compass distances; leave arcs visible.
Perpendicular bisector: equidistant from two points.
Angle bisector: equidistant from two lines.
Fixed distance from a point is a circle; from a segment, a running-track shape.
Combine loci and shade the region satisfying all conditions.

Exam-style questions

Practise it the way it is examined

3 original questions · 7 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 31 mark

Describe the locus of points that are the same distance from two straight lines that meet at a point.[1]

Question 2Grade 53 marks

A goat is tied to a corner post of a large rectangular field by a rope 7 m long.

Work out the area of the field the goat can reach. Give your answer to 1 decimal place.[3]

Question 3Grade 63 marks

Describe how to construct, using only a ruler and compasses, the perpendicular from a point $P$ to a line $L$ .[3]

Explore further

OCR J560 official specification (opens in a new tab)
Check the assessed scope and practical or mathematical requirements.
Read or explore this topic (opens in a new tab)
An external explanation or interactive activity to use alongside the lesson. It may include extension material.

Return to this tomorrow

Close the notes and answer from memory. Revisit any explanation you cannot yet reconstruct.

Points equidistant from two intersecting lines lie on:
On a map with scale 1 cm : 5 km, what radius shows 12 km?
The region "closer to $P$ than to $Q$ " is bounded by:
The locus of points 2 m from a straight fence 10 m long is:

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 4 lessons

Constructions & Loci

With only a ruler and a pair of compasses you can bisect an angle, draw a perpendicular exactly through the middle of a line, and mark out every point a given distance from a landmark. These

\textbf{constructions}

solve

\textbf{loci}

problems: where can a goat tethered to a post graze? Which part of a park is closer to the café than the gate?

45 min Angles & Geometric Reasoning

Watch, pause, try

Video by corbettmaths. The video’s own board label is retained; follow the specification reference and scope notes on this lesson for assessed content.

Viewing focus: Construct the perpendicular bisector of a line segment

After watching, attempt these questions before opening the answers. You can also use the written lesson below.

After watching 1: The locus of points 3 cm from a point is:

Show answer and reasoning

A circle of radius 3 cm. All points at a fixed distance from a centre.

After watching 2: Points equidistant from $A$ and $B$ lie on:

Show answer and reasoning

The perpendicular bisector of $AB$ . By construction.

Practise it the way it is examined

Question 1Grade 31 mark

Describe the locus of points that are the same distance from two straight lines that meet at a point.[1]

Question 2Grade 53 marks

A goat is tied to a corner post of a large rectangular field by a rope 7 m long.

Work out the area of the field the goat can reach. Give your answer to 1 decimal place.[3]

Question 3Grade 63 marks

Describe how to construct, using only a ruler and compasses, the perpendicular from a point $P$ to a line $L$ .[3]

Return to this tomorrow

Close the notes and answer from memory. Revisit any explanation you cannot yet reconstruct.

Points equidistant from two intersecting lines lie on:

On a map with scale 1 cm : 5 km, what radius shows 12 km?

The region "closer to

P

than to

Q

" is bounded by:

The locus of points 2 m from a straight fence 10 m long is: