Explaining what stays unchanged | Year 7

01 / LEARN

The idea, step by step

Invariant means unchanged

Translations, reflections and rotations are rigid transformations: the shape moves without stretching. Corresponding lengths and angles remain equal, so the image and original are congruent. Their perimeters and areas are equal too.

Position usually changes, so the coordinates need not stay the same. Reflection reverses the clockwise order of labelled vertices. Translation and rotation preserve that order. A flipped shape is still congruent even though its handedness changes.

Use measurements to test a claim

If a rectangle is 3 units wide and 2 units high, any rigid image still has perimeter $2 (3 + 2) = 10$ units and area $3 \times 2 = 6$ square units. A drawing on another part of the grid is not automatically a different-sized rectangle.

A rule that doubles every length is an enlargement, not one of these rigid transformations. The new shape can have the same angles while having a different perimeter and area. One preserved property alone is not enough to prove a rigid transformation.

A common trap

Saying a reflection changes the area because the shape faces the other way.

Instead: A reflection changes handedness, not size. Corresponding lengths, angles and area remain equal.

02 / WORKED EXAMPLES

See the thinking

Example 1

A rectangle slides

A 3 cm by 2 cm rectangle is translated. Find the image perimeter and area.

Translation preserves both side lengths.
The perimeter is $2 (3 + 2) = 10$ cm.
The area is $3 \times 2 = 6$ square centimetres.

Perimeter 10 cm; area

6 {cm}^{2}

Example 2

A reflected right triangle

A right triangle has perpendicular sides 3 cm and 2 cm. It is reflected in a line. What happens to its right angle, area and vertex order?

The right angle remains 90° because reflection preserves angles.
The area remains $\frac{1}{2} \times 3 \times 2 = 3 {cm}^{2}$ .
The clockwise/anticlockwise order reverses under reflection.

Right angle and area are unchanged; vertex order is reversed.

Check your understanding

A triangle of area $8 {cm}^{2}$ is rotated. What is the image area?

Show answer

$8 {cm}^{2}$

Rotation is rigid, so it preserves area.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A 5 cm by 2 cm rectangle is reflected. Find its image perimeter and area.
2 marks
Which properties stay unchanged under a translation: coordinates, side lengths, angle sizes, area?
2 marks
A triangle is rotated 90°. Does its clockwise vertex order reverse? Explain.
2 marks
A rule maps a 2 cm side to a 4 cm side. Could it be a translation, reflection or rotation? Explain.
2 marks
A right triangle has perpendicular sides 4 units and 3 units. It is reflected in the y-axis. Find its area and state whether it is congruent to the original.
3 marks
A student claims two shapes must be congruent because they have equal area. Is this enough? Give a counterexample.
3 marks

Read the worked solutions

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

A 5 cm by 2 cm rectangle is reflected. Find its image perimeter and area.
2 marks
The side lengths stay 5 and 2.
$2 (5 + 2) = 14$ and $5 \times 2 = 10$ .
Perimeter 14 cm; area $10 {cm}^{2}$ .
Which properties stay unchanged under a translation: coordinates, side lengths, angle sizes, area?
2 marks
Coordinates generally change because the shape changes position.
Translation does not stretch the shape.
Side lengths, angle sizes and area.
A triangle is rotated 90°. Does its clockwise vertex order reverse? Explain.
2 marks
All vertices turn together about the centre; there is no mirror reversal.
No. A rotation preserves clockwise/anticlockwise order.
A rule maps a 2 cm side to a 4 cm side. Could it be a translation, reflection or rotation? Explain.
2 marks
The side has doubled, so the rule includes a size change.
No. Each of those transformations preserves side lengths.
A right triangle has perpendicular sides 4 units and 3 units. It is reflected in the y-axis. Find its area and state whether it is congruent to the original.
3 marks
$\frac{1}{2} \times 4 \times 3 = 6$ .
Reflection preserves all corresponding lengths and angles.
Area 6 square units; congruent.
A student claims two shapes must be congruent because they have equal area. Is this enough? Give a counterexample.
3 marks
Equal area alone does not establish equal corresponding lengths and angles.
No. A 1 by 6 rectangle and a 2 by 3 rectangle both have area 6 but different side lengths.

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Curriculum reference & lesson scope

Australian Curriculum v9 / South Australian Curriculum

Translations; reflections in coordinate axes; rotations about a point; recording transformations with coordinates; sequences of transformations; explaining what stays unchanged.

Official curriculum source ↗

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