Isometries
Senior 2 · Mathematics
Isometries
Lesson 5 of 5
20 min
A carpenter who cuts eight identical chair legs from one template, a tiler laying a school floor, and a designer repeating a motif along a kitenge all rely on the same fact: an isometry never changes size or shape. In this lesson you gather the properties that all four isometries share, find out what happens when one is applied twice, and use isometries to decide when two figures are congruent.
Central symmetry, reflection, rotation and translation all keep:
So the image of any figure is congruent to it: same shape, same size. In fact, two figures are congruent exactly when one can be moved onto the other by a sequence of translations, rotations and reflections.
| Isometry | Fixed points | Orientation |
|---|---|---|
| Central symmetry | the centre only | kept |
| Rotation (not 0° or 360°) | the centre only | kept |
| Translation (v not zero) | none | kept |
| Reflection | every point of the mirror line | reversed |
Apply the same transformation a second time to the image:
Worked example 1
Translate P(3, 2) twice by 1−4.
(3, 2) → (4, −2) → (5, −6). In one step: 2 × 1−4 = 2−8, and (3 + 2, 2 − 8) = (5, −6).
Worked example 2
Rotate P(3, 2) twice through +90° about the origin.
(3, 2) → (−2, 3) → (−3, −2). That is (−x, −y): a half-turn, the central symmetry about O.
Worked example 3
Triangle A(1, 1), B(3, 1), C(1, 2) is mapped to A′(−1, −1), B′(−3, −1), C′(−1, −2). Name the isometry.
Every point follows (x, y) → (−x, −y). This is the central symmetry about O, which is also the rotation through 180° about O.
Worked example 4
Is it possible that a reflection maps triangle PQR onto triangle P′Q′R′ if PQR is labelled clockwise and P′Q′R′ is labelled clockwise too?
No. A reflection reverses orientation, so the image would be labelled anticlockwise. A rotation, a translation or a central symmetry could do it.
Tip
Reflection is the odd one out: it is the only one of the four that reverses orientation, and the only one with a whole line of fixed points. When you must name an isometry, check orientation first; it rules out half the options at once.
Reflect (2, 7) in the x-axis, then reflect the image in the x-axis again.
Solution: (2, 7) → (2, −7) → (2, 7). It is back where it started.
Translate (1, 1) three times by 32.
Solution: 3 × 32 = 96, so the image is (10, 7).
An isometry maps (4, 1) to (−1, 4) and (2, 3) to (−3, 2). Which one is it?
Solution: both follow (x, y) → (−y, x): the rotation through +90° about the origin.
A triangle of area 12 cm² is reflected in a line. What is the area of the image?
Solution: 12 cm². Isometries keep areas.
Which isometry keeps orientation and has exactly one fixed point?
Solution: a rotation (through an angle that is not 0° or 360°); the central symmetry is the special case of 180°.
Remember
Sources and credits
The Rwandan examples, practice items and exercises are written by Ganzaa.
