Geometry · Grade 8
Transformations, Congruence and Similarity
Transformations move or resize shapes. They help describe symmetry, congruence and similarity precisely.
Learning time: 20–25 minutes · Level: Developing
Learning goals
- Understand the key terms and concepts.
- Apply a reliable method to worked and new problems.
- Check your reasoning and communicate your answer.
1. Translations
A translation slides every point the same distance in the same direction. Shape, size and orientation stay the same.
2. Reflections and rotations
A reflection flips a shape across a line called the mirror line. A rotation turns a shape around a fixed centre by a specified angle and direction.
3. Congruent figures
Congruent figures have the same shape and size. One can be mapped onto the other using rigid transformations: translations, rotations and reflections.
4. Similar figures
Similar figures have the same shape but may have different sizes. Corresponding angles are equal and corresponding side lengths are in the same ratio.
5. Scale factors
The scale factor is image length divided by original length. A factor greater than 1 enlarges; a factor between 0 and 1 reduces. Areas scale by the square of the factor.
Track coordinates and scale factors
A translation slides a figure without turning it; a reflection flips it across a line; and a rotation turns it around a fixed centre. These rigid transformations preserve lengths and angles, so the image is congruent to the original. A dilation changes size by a scale factor. It preserves angle measures and shape, so the figures are similar.
Worked reasoning: A point (2, 1) is translated 3 units right and 2 units down. Add 3 to the x-coordinate and subtract 2 from the y-coordinate: (2 + 3, 1 − 2) = (5, −1). The movement does not change the shape or size of a figure.
If every length is multiplied by a scale factor of 3, the area is multiplied by 3² = 9. A common mistake is to multiply area by 3 rather than by 9.
- Apply coordinate changes consistently to every vertex.
- Check whether a transformation preserves size or changes it.
- Use the square of the scale factor for area comparisons.
Worked examples
- A point (2, 1) translated by (3, −2) becomes (5, −1).
- A 90° anticlockwise rotation about the origin maps (x, y) to (−y, x).
- A side of 4 cm enlarged by scale factor 2.5 becomes 10 cm.
- If similar shapes have side scale factor 3, their area scale factor is 9.
Independent practice
Attempt each problem before checking the worked answer.
- Translate (1, 4) by (2, −3). Answer: (3, 1).
- Reflect (−3, 5) in the y-axis. Answer: (3, 5).
- Rotate (2, 1) 90° anticlockwise about the origin. Answer: (−1, 2).
- Are two squares with side lengths 3 cm and 7 cm congruent? Answer: No; they are similar but different sizes.
- A 6 cm side is enlarged by factor 4. Answer: 24 cm.
- A shape is reduced from 10 cm to 5 cm. Answer: scale factor 0.5.
- Similar figures have side scale factor 2. What is area scale factor? Answer: 4.
- If corresponding angles are equal and corresponding sides proportional, what is the relationship? Answer: Similarity.
Knowledge check
Choose one answer for each question, then check your answers for explanations.
Printable worksheet and answer key
Use the independent-practice list above as your worksheet. Show working, label units where appropriate, and review the answers after attempting all questions.
Study reminder
Write the rule or method used beside each solution. This helps identify and correct mistakes.