Isometries
Senior 2 · Mathematics
Isometries
Lesson 1 of 5
22 min
Look closely at an imigongo panel, the black-and-white geometric art from eastern Rwanda. One motif is copied again and again: slid along, flipped over, or turned round. Floor tiles, kitenge prints and woven baskets are built the same way. Each copy has exactly the same size and shape as the original; only its position has changed.
A movement of the plane that never changes lengths is called an isometry (from Greek: "equal measure"). Engineers use isometries to check that machine parts are identical, designers use them to build patterns, and in this unit you will use them to describe exactly where every point of a shape goes.
A transformation moves every point P of the plane to a new point P′, its image. We write P → P′ and read it "P maps to P′". The starting figure is the object; the moved figure is the image. Points that match up, such as A and A′, are called corresponding points.
A transformation is an isometry (also called a rigid or congruent transformation) if, for every pair of points A and B, A′B′ = AB. There is no bending and no stretching. In this unit you will study four isometries: central symmetry, reflection, rotation and translation. An enlargement is not an isometry, because it changes lengths.
A central symmetry with centre O (also called a half-turn, or a point reflection) maps each point A to the point A′ such that O is the midpoint of AA′.
To construct A′: draw the line from A through O, and continue it beyond O by the same length OA.
Worked example 1
Find the image of triangle A(3, 1), B(5, 4), C(1, 3) under central symmetry about the origin.
Change every sign: A′(−3, −1), B′(−5, −4), C′(−1, −3).
Figure 1. Central symmetry about O: each dashed segment joins a point to its image and has O as its midpoint. Diagram: Ganzaa.
Worked example 2
Find the image of P(6, −1) under central symmetry about C(2, 3).
P′ = (2 × 2 − 6, 2 × 3 − ( − 1)) = (4 − 6, 6 + 1) = ( − 2, 7).
Check: the midpoint of PP′ is ((6 − 2) ÷ 2, ( − 1 + 7) ÷ 2) = (2, 3), which is C.
A figure that is its own image under a central symmetry has point symmetry: a parallelogram (about the point where its diagonals cross), and the letters S, N and Z.
Tip
The centre is the midpoint. Image = 2 × centre − point. For centre (2, 3) and point (6, −1): (4 − 6, 6 + 1) = ( − 2, 7). Then check with the midpoint formula; it takes ten seconds and catches every sign slip.
Find the image of (−4, 2) under central symmetry about the origin.
Solution: (4, −2).
Find the image of (1, 5) under central symmetry about C(3, 0).
Solution: (2 × 3 − 1, 2 × 0 − 5) = (5, −5).
A central symmetry maps A(1, −1) to A′(7, 3). Find its centre.
Solution: the centre is the midpoint of AA′: ((1 + 7) ÷ 2, ( − 1 + 3) ÷ 2) = (4, 1).
Which of these have point symmetry: the letter Z, the letter A, a parallelogram, an isosceles triangle?
Solution: the letter Z and the parallelogram. Turning A or an isosceles triangle through 180° turns it upside down.
Is an enlargement with scale factor 2 an isometry?
Solution: no. Every length is doubled, so A′B′ ≠ AB.
Remember
Sources and credits
The Rwandan examples, practice items and exercises are written by Ganzaa.
This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.
