The idea, step by step
Movements that preserve size
Translations, rotations and reflections are rigid transformations: corresponding lengths and angles stay unchanged. A translation adds the same horizontal and vertical change to every point. Reflection in the y axis changes the sign of x but leaves y unchanged.
The two triangles shown are congruent. The reflection reverses orientation, but each 2-unit leg remains 2 units. You can check coordinates as well as matching corresponding parts.
Enlargement has a different effect
An enlargement with centre the origin multiplies both coordinates by the scale factor. Multiplying every coordinate by 3 creates sides three times as long, while corresponding angles stay unchanged. The image is similar, but it is not congruent unless the size factor is 1.
A digital image’s width and height must be multiplied by the same factor to keep its proportions. Stretching only the width is not a similarity transformation.
A common trap
Changing only one coordinate by a scale factor and still calling the result similar.
Instead: An enlargement about the origin multiplies both coordinates by the same factor.
See the thinking
Translate every vertex
Triangle ABC has A(1,2), B(4,2), C(1,4). Translate it 2 units right and 1 unit down. Find the image vertices.
- The translation rule is .
- Apply it to each vertex: A′(3,1), B′(6,1), C′(3,3).
- Each point moved the same distance and direction, so the triangles are congruent.
Compare an enlargement
A rectangle has vertices (0,0), (2,0), (2,1), (0,1). Enlarge it by factor 3 about the origin.
- Multiply each coordinate by 3.
- The image vertices are (0,0), (6,0), (6,3), (0,3).
- Its width and height are both tripled, so it is similar but not congruent to the original.
Check your understanding
Reflect point (4,−2) in the y axis.
Show answer
(−4,−2).
Change the sign of x and keep y unchanged.
Your turn
6 questions · 13 marks · Work at your own pace.
Read the questions online
- Translate point (−2,3) 5 units right and 2 units down.2 marks
- Reflect point (−3,2) in the x axis.2 marks
- Enlarge point (2,−1) by factor 4 about the origin.2 marks
- A triangle is rotated 90° without changing its size. Is the image congruent, similar, both or neither?2 marks
- An image is stretched from width 4 cm and height 3 cm to width 8 cm and height 3 cm. Is it similar? Explain.2 marks
- Triangle ABC has A(0,0), B(2,0), C(0,1). Plot its image after translation 1 unit right and 2 units up.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Translate point (−2,3) 5 units right and 2 units down.2 marks
and .
(3,1). - Reflect point (−3,2) in the x axis.2 marks
The x coordinate stays −3; the y coordinate changes sign.
(−3,−2). - Enlarge point (2,−1) by factor 4 about the origin.2 marks
Multiply both coordinates by 4.
(8,−4). - A triangle is rotated 90° without changing its size. Is the image congruent, similar, both or neither?2 marks
Rotation preserves lengths and angles.
Congruence is similarity with scale factor 1.
Both. - An image is stretched from width 4 cm and height 3 cm to width 8 cm and height 3 cm. Is it similar? Explain.2 marks
The width factor is 2 but the height factor is 1.
Corresponding dimensions do not share one factor.
No. - Triangle ABC has A(0,0), B(2,0), C(0,1). Plot its image after translation 1 unit right and 2 units up.3 marks
Apply to all three vertices.
Plot and join the three image points.
A′(1,2), B′(3,2), C′(1,3).
Curriculum reference & lesson scope
Australian Curriculum v9 / SA Curriculum
Congruent shapes and matching parts; triangle congruence tests; similar shapes and corresponding sides; triangle similarity; properties of quadrilaterals established using triangles; testing congruence/similarity with transformations; decision-tree algorithms and digital geometry.
Official curriculum source ↗