Invariant and changing properties | Year 9

01 / LEARN

Understand the idea.

Know what stays unchanged

A property is invariant if a transformation leaves it unchanged. Translation, rotation and reflection preserve all lengths and angles, so they also preserve perimeter and area. Their images have the same shape and size as the original.

Enlargement preserves corresponding angles and length ratios, but generally changes size. A positive enlargement preserves clockwise vertex order; a reflection reverses that order. Position is usually changed, although points on a mirror line remain fixed.

Scale lengths and areas correctly

For positive scale factor $k$ , each side and the total perimeter are multiplied by $k$ . Area is multiplied by $k^{2}$ : a rectangle changes from $l w$ to $(k l) (k w) = k^{2} l w$ , because both dimensions are scaled.

If you know the area multiplier, take its positive square root to recover the positive length factor. Compare corresponding quantities in the same units and distinguish a length multiplier from an area multiplier.

P^{'} = k P, A^{'} = k^{2} A

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Rectangle enlargement

A rectangle is 3 cm by 5 cm. Enlarge by factor 2 and find the new perimeter and area.

Worked solution

1
Scale both rectangle dimensions.
$\begin{aligned} 3 \times 2 & = 6 cm \\ 5 \times 2 & = 10 cm \end{aligned}$
The factor $2$ applies to every length.
2
Calculate the new perimeter.
$P^{'} = 2 (6 + 10) = 32 cm$
A rectangle has two pairs of equal sides.
3
Calculate the new area.
$A^{'} = 6 \times 10 = 60 {cm}^{2}$
Area multiplies the two new perpendicular dimensions.
4
Check the two multipliers.
$\begin{aligned} \frac{P^{'}}{P} & = \frac{32}{16} = 2 \\ \frac{A^{'}}{A} & = \frac{60}{15} = 4 = 2^{2} \end{aligned}$
Perimeter doubles, while area increases by the square of the factor.

Answer

32 cm and 60 cm².

EXAMPLE 02

A non-integer factor

A right triangle has sides 3, 4 and 5 cm. Enlarge by factor 1.5. Find the image sides and the angle between its two shorter sides.

Worked solution

1
Scale the shorter sides.
$\begin{aligned} 3 (1.5) & = 4.5 cm \\ 4 (1.5) & = 6 cm \end{aligned}$
Each corresponding length is multiplied by $1.5$ .
2
Scale the longest side.
$5 (1.5) = 7.5 cm$
The hypotenuse uses the same length factor as the other sides.
3
Identify the corresponding angle.
$θ^{'} = θ = 90^{\circ}$
Enlargement preserves angles, so the right angle between the two shorter sides remains a right angle.
4
Check the common side ratio.
$\frac{4.5}{3} = \frac{6}{4} = \frac{7.5}{5} = 1.5$
All three corresponding sides have the required scale factor.

Answer

Sides 4.5, 6, 7.5 cm; the matching right angle is 90°.

03 / YOUR TURN

Try it yourself.

Check your understanding

Under enlargement by factor 3, what happens to a 50° angle?

Show answer & reasoning

It stays 50°.

The angle remains $50^{\circ}$ . Enlargement multiplies corresponding lengths by $3$ , while corresponding angles are invariant.

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Name two properties preserved by translation.
2 marks
A side is 2.5 cm. Find its image length under factor 4.
2 marks
A perimeter is 22 cm. Find the image perimeter under factor 1.5.
2 marks
An area is 18 cm². Find the image area under factor 3.
2 marks
Similar figures have areas 25 cm² and 100 cm². Find the positive length factor from small to large.
2 marks
A triangle is reflected. What happens to its clockwise vertex order and area? Explain.
3 marks

Read the worked solutions

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

Name two properties preserved by translation.
2 marks
Every point moves by the same displacement, without changing shape or size.
For example, side lengths and angles.
A side is 2.5 cm. Find its image length under factor 4.
2 marks
$4 \times 2.5 = 10$ .
10 cm
A perimeter is 22 cm. Find the image perimeter under factor 1.5.
2 marks
Perimeter scales by 1.5.
$22 \times 1.5 = 33$ .
33 cm
An area is 18 cm². Find the image area under factor 3.
2 marks
Area factor is $3^{2} = 9$ .
$18 \times 9 = 162$ .
162 cm²
Similar figures have areas 25 cm² and 100 cm². Find the positive length factor from small to large.
2 marks
Area factor is $100 \div 25 = 4$ .
Length factor is $\sqrt{4} = 2$ .
$2$
A triangle is reflected. What happens to its clockwise vertex order and area? Explain.
3 marks
Reflection reverses orientation.
It preserves shape and size, so the triangles are congruent.
The order reverses; area stays the same.

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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.

Know what stays unchanged

Scale lengths and areas correctly

See the method in action.

Rectangle enlargement

Scale both rectangle dimensions.

Calculate the new perimeter.

Calculate the new area.

Check the two multipliers.

A non-integer factor

Scale the shorter sides.

Scale the longest side.

Identify the corresponding angle.

Check the common side ratio.

Try it yourself.

Check your understanding

Keep practising