KS3 Maths · Year 7 · Chapter 2: Geometry · Lesson 5

Constructions

Year 7About 80 minutesNo calculator45 marks

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Quick questions to start.

A1
Needed today
An angle of $8 4^{\circ}$ is cut in half. How big is each half?
[1 mark]
Answer
A2
Needed today
A line is $9.4$ cm long. How far is its mid-point from each end?
[1 mark]
Answer
A3
Last lesson
A $3$ cm by $5$ cm rectangle is enlarged by scale factor $4$ . How long is its longer side now?
[1 mark]
Answer
A4
From chapter 1
Work out $- 8 + 3 \times (- 2)$ .
[1 mark]
Answer

Example, then your turn

Year 7

Use a ruler and compasses, and leave all your arcs. Your teacher works through each example with you. Then try the one beside it.

Example 1

On the grid, construct the perpendicular bisector of $A B$ . Write down the mid-point of $A B$ , and where the bisector crosses the $x$ -axis.

Hint Set your compasses to more than half of

A B

. Draw arcs from

A

and

B

, above and below, with the same setting.

B1
On the grid, construct the perpendicular bisector of $A B$ . Write down the mid-point of $A B$ , and where the bisector crosses $y = 1$ .
[3 marks]
Answer

Example 2

On the diagram, construct the bisector of angle $A B C$ . Measure the two angles you have made.

Hint Draw an arc from

B

that cuts both arms. From those two points draw two equal arcs that cross. Join

B

to the crossing.

B2
On the diagram, construct the bisector of angle $A B C$ . Measure the two angles you have made.
[3 marks]
Answer

Example 3

Construct the triangle $A B C$ with $B C = 5$ cm, angle $A B C = 9 0^{\circ}$ and $A C = 6.5$ cm. Measure $A B$ .

Not drawn accurately

Hint Draw

B C

longer than you need. Construct the perpendicular at

B

. Then set your compasses to

6.5

cm and draw an arc from

C

B3
Construct the triangle $P QR$ with $QR = 4.5$ cm, angle $P QR = 9 0^{\circ}$ and $P R = 5.8$ cm. Measure $P Q$ .
[3 marks]
Not drawn accurately
Answer

Practice

Year 7

C1
$A$ and $C$ are opposite corners of the rhombus $A B C D$ . On the grid, construct the perpendicular bisector of $A C$ . $B$ is on the line $y = 4$ and $D$ is on the $x$ -axis. Write down $B$ and $D$ .
[3 marks]
Answer
C2
Angle $A B C$ is bisected. Then the half next to $B C$ is bisected again. The smallest angle made is $1 9^{\circ}$ . What was the size of angle $A B C$ ?
[3 marks]
Not drawn accurately
Answer

C3
$P$ is on the perpendicular bisector of $A B$ . $P A = 6.3$ cm. Write down the length of $P B$ . Give a reason.
[2 marks]
Not drawn accurately
Answer

Exam-style questions

GCSE Foundation

D1
Priya wants to construct the perpendicular bisector of a line $A B$ that is $9$ cm long. She says, "I will set my compasses to $4$ cm for the arcs from $A$ and from $B$ ."
Explain why her construction will not work. Write down a radius she could use. How far from $A$ will the bisector cross $A B$ , and at what angle?
[5 marks]

D2
The grid shows a straight road and a house $H$ . On the grid, construct the shortest path from $H$ to the road. Write down the coordinates of the point where the path meets the road.
[4 marks]
Answer
D3
Two straight hedges meet at $V$ . A path from $V$ is always the same distance from both hedges.
(a)
On the diagram, use a ruler and compasses to construct the path.
[2 marks]
(b)
Work out the angle between the path and each hedge.
[1 mark]
Answer
(c)
A tree on the path is $4.6$ m from one hedge. How far is it from the other hedge? Give a reason.
[2 marks]
Answer
Total for D3: 5 marks
D4
A ladder $5$ m long leans against a vertical wall. Its foot is on level ground, $1.4$ m from the wall. Use a scale of $1$ cm to $1$ m. Construct an accurate scale drawing in the space. Then work out how high up the wall the ladder reaches, and measure the angle between the ladder and the ground.
[5 marks]
Not drawn accurately
Answer

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A circle passes through the points $A$ , $B$ and $C$ on the grid. Use constructions to find the coordinates of the centre of the circle. Then write down the radius of the circle, in grid squares.
[5 marks]
Hint 1 · What to try
The centre is the same distance from $A$ as from $B$ . Which line holds every point like that?
Hint 2 · The first line
Construct the perpendicular bisector of $A B$ and the perpendicular bisector of $B C$ . The centre is where they cross.
Hint 3 · The full method
They cross at $(4, 0)$ . $B$ is $5$ squares above $(4, 0)$ , so the radius is $5$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can construct the perpendicular bisector of a line and the bisector of an angle.
I can construct a perpendicular from a point to a line and at a point on a line.
I can construct a right-angled triangle and measure its sides and angles.

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