What is a transformation?
A transformation is a change made to a shape on a plane. The starting shape is called the object, and the new shape produced is called the image. We usually label the object with letters such as A, B, C and its image with A′, B′, C′ (read as A prime).
Some transformations keep the shape and size of the object exactly the same, while others do not. When the image has the same shape and size as the object, we say the two shapes are congruent.
Isometric and non-isometric
An isometric transformation (an isometry) preserves distance. The object and image are congruent — only the position or orientation may change. The three isometric transformations in Form 2 are translation, reflection and rotation.
Key idea / 要点
Isometry keeps lengths and angles unchanged, so object and image are always congruent. Enlargement changes size, so it is NOT an isometry.
A transformation that changes size, such as an enlargement, is not isometric because the image is bigger or smaller than the object.
Worked example / 例题
Triangle A has sides 3 cm, 4 cm and 5 cm. After a transformation, its image A′ has sides 3 cm, 4 cm and 5 cm in a new position. Is this an isometry?
The side lengths are unchanged (3, 4, 5), so the image is congruent to the object. Therefore the transformation IS an isometry.
Remember / 记住
- Object → image; label the image with primes (A′).
- Isometry keeps shape and size the same (congruent).
- Translation, reflection and rotation are isometries.
- Enlargement is NOT an isometry.