The idea, step by step
Know what stays unchanged
Translation, rotation and reflection preserve lengths, angles, perimeter and area. Enlargement preserves angles and corresponding length ratios, so its image is similar, but size normally changes.
A positive enlargement preserves clockwise vertex order. Reflection reverses it. A point on a mirror line stays fixed even though other points move, so position is not generally invariant.
Scale lengths and areas correctly
If every length scales by k, perimeter also scales by k. Area scales by k² because two dimensions change. Confirm this by multiplying the new dimensions of a rectangle.
When given an area ratio, its positive square root gives the positive length factor. Always compare corresponding measures in the same units.
A common trap
Doubling lengths doubles area.
Instead: Both dimensions double, so area is multiplied by 2²=4.
See the thinking
Rectangle enlargement
A rectangle is 3 cm by 5 cm. Enlarge by factor 2 and find the new perimeter and area.
- New dimensions are 6 cm by 10 cm.
- Perimeter 2(6+10)=32 cm, twice the original 16 cm.
- Area 6×10=60 cm², four times the original 15 cm².
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.
- Multiply each side by 1.5 to obtain 4.5, 6 and 7.5 cm.
- The two shorter sides originally meet at 90°.
- The corresponding angle stays 90° and the side ratio is unchanged.
Check your understanding
Under enlargement by factor 3, what happens to a 50° angle?
Show answer
It stays 50°.
Corresponding angles are invariant under enlargement.
Your turn
6 questions · 13 marks · Work at your own pace.
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
4×2.5=10.
10 cm - A perimeter is 22 cm. Find the image perimeter under factor 1.5.2 marks
Perimeter scales by 1.5.
22×1.5=33.
33 cm - An area is 18 cm². Find the image area under factor 3.2 marks
Area factor is 3²=9.
18×9=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÷25=4.
Length factor is √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.
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 ↗