What makes a transformation an isometry?
An isometry is a transformation that preserves distance. This means every length in the object is exactly equal to the matching length in the image. Because lengths and angles are unchanged, the object and its image are always congruent.
The three transformations you have met — translation, reflection and rotation — are all isometries. Each moves a shape to a new position without stretching, shrinking or distorting it.
Key idea / 要点
Isometry ⇒ object and image are congruent. To prove an isometry, check that a side length in the object equals the matching side length in the image.
Comparing the three
| Transformation | Needs |
|---|---|
| Translation | a vector |
| Reflection | an axis |
| Rotation | centre, angle, direction |
Worked example / 例题
Triangle ABC has AB = 6 cm. After a rotation about O, its image is A′B′C′ with A′B′ = 6 cm. Explain why this is an isometry.
Because AB = A′B′ = 6 cm, the length is unchanged. A rotation preserves all distances, so the image is congruent to the object — it is an isometry.
An enlargement is NOT an isometry because it changes side lengths, so the image is not congruent to the object.
Remember / 记住
- Isometry preserves distance and angle.
- Translation, reflection, rotation are all isometries.
- Object and image are congruent.
- Enlargement is not an isometry.