Geometric constructions | Year 9

01 / LEARN

Understand the idea.

Construct a perpendicular bisector

A perpendicular bisector meets a segment at $90^{\circ}$ and divides it into equal halves. Open the compass to a radius greater than half the segment length. With unchanged width, draw arcs from each endpoint to create two intersections.

Join those intersections. Each lies equally far from both endpoints because the compass radii match, so their line is the perpendicular bisector. Equal radii are essential; a radius equal to half the segment gives only one touching point.

Bisect an angle or construct a triangle

To bisect an angle, draw an arc centred at the vertex, meeting both arms. From these equal-distance points, draw equal-radius arcs meeting inside the angle. Joining that crossing to the vertex creates two congruent triangles and hence equal angles.

For an equilateral triangle on a segment, draw circles centred at its endpoints with radius equal to the segment. Join either intersection to both endpoints. All three sides are equal radii, so the construction determines the triangle without estimating its angles.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Bisect a segment

Construct the perpendicular bisector of a 6 cm segment AB.

Worked solution

1
Draw the segment and choose the compass width.
$A B = 6 cm, r > 3 cm$
The radius must exceed half the segment length to create two intersections.
2
Draw matching arcs from both endpoints.
$A U = B U = r, A V = B V = r$
Keep the compass width fixed while drawing arcs above and below the segment; name the crossings $U$ and $V$ .
3
Join the two arc crossings.
$Draw line U V$
Both crossings are equidistant from $A$ and $B$ , so they lie on the perpendicular bisector.
4
State the constructed properties.
$A M = M B = 3 cm, U V ⟂ A B$
At $M = U V \cap A B$ , the line halves the segment and meets it at a right angle.

Answer

A perpendicular line through the midpoint.

EXAMPLE 02

Bisect an angle

An angle is 80°. Describe its bisector construction and find the new angles.

Worked solution

1
Mark equal distances on the angle arms.
$O D = O E$
Let $O$ be the vertex; one arc centred at $O$ meets the arms at $D$ and $E$ .
2
Make an interior crossing with equal arcs.
$D X = E X$
Use the same compass width at $D$ and $E$ so their arcs meet at $X$ inside the angle.
3
Join the crossing to the vertex.
$O D = O E, D X = E X, O X = O X$
Triangles $O D X$ and $O E X$ are congruent by three equal corresponding sides, so $O X$ bisects the angle.
4
Calculate the equal angles.
$80^{\circ} \div 2 = 40^{\circ}$
The bisector divides the original angle into two equal parts: $40^{\circ}$ and $40^{\circ}$ .

Answer

40° and 40°.

03 / YOUR TURN

Try it yourself.

Check your understanding

Is radius 3 cm enough for the usual bisector construction on an 8 cm segment?

Show answer & reasoning

No.

Half of $8 cm$ is $4 cm$ . A radius of $3 cm$ gives two circles with no intersection, so choose the same radius greater than $4 cm$ at both ends.

Keep practising

6 worksheet questions · 19 marks

Open worksheet Worked solutions

Read the questions online

Draw an 8 cm segment and construct its perpendicular bisector. State a suitable radius.
4 marks
Construct the bisector of a $60^{\circ}$ angle. State the new angles.
3 marks
Construct an equilateral triangle with side length 5 cm. Explain why the sides are equal.
4 marks
A point $P$ is equally far from $A$ and $B$ . On which line must $P$ lie?
2 marks
A student changes compass width between the arcs from $A$ and $B$ . Why does the usual proof fail?
3 marks
Two circles have radius 4 cm and centres 8 cm apart. Why are they unsuitable for the usual two-crossing construction?
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Draw an 8 cm segment and construct its perpendicular bisector. State a suitable radius.
4 marks
Use the same radius greater than 4 cm at each endpoint.
Join the two crossings.
For example, radius 5 cm; the line crosses at the 4 cm midpoint at $90^{\circ}$ .
Construct the bisector of a $60^{\circ}$ angle. State the new angles.
3 marks
Use equal-distance arm marks and equal-radius crossing arcs.
$60^{\circ} \div 2 = 30^{\circ}$ .
$30^{\circ}$ and $30^{\circ}$ .
Construct an equilateral triangle with side length 5 cm. Explain why the sides are equal.
4 marks
Draw $A B = 5$ cm.
Draw circles of radius 5 cm centred at $A$ and $B$ .
Join an intersection $C$ to $A$ and $B$ . Both new sides are radii.
Each side is 5 cm.
A point $P$ is equally far from $A$ and $B$ . On which line must $P$ lie?
2 marks
The locus of points equally far from the endpoints is their perpendicular bisector.
The perpendicular bisector of $A B$ .
A student changes compass width between the arcs from $A$ and $B$ . Why does the usual proof fail?
3 marks
Different radii represent different distances.
The construction needs equal distances to both endpoints.
The crossings are not guaranteed to be equally far from $A$ and $B$ .
Two circles have radius 4 cm and centres 8 cm apart. Why are they unsuitable for the usual two-crossing construction?
3 marks
The radius is exactly half the centre distance.
Two distinct crossings are needed to determine the joining line.
They touch at one point only.

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Curriculum reference & lesson scope

Australian Curriculum v9 / South Australian Curriculum

Angle reasoning consolidation; enlargements using coordinates/digital geometry; invariant and changing properties; geometric constructions; sequences of steps and decisions; testing/refining construction algorithms; explaining geometric reasoning.

Official curriculum source ↗

In this lesson

Understand the idea.

Construct a perpendicular bisector

Bisect an angle or construct a triangle

See the method in action.

Bisect a segment

Draw the segment and choose the compass width.

Draw matching arcs from both endpoints.

Join the two arc crossings.

State the constructed properties.

Bisect an angle

Mark equal distances on the angle arms.

Make an interior crossing with equal arcs.

Join the crossing to the vertex.

Calculate the equal angles.

Try it yourself.

Check your understanding

Keep practising