Year 10 Extended · Unit 5 Solids, similarity and transformations · Lesson 6 of 6

Every move is made of mirrors

Two reflections make a slide or a turn, and the size of that slide or turn depends only on the mirrors. This lesson proves it, shows that three mirrors can do any isometry, and asks when the order of two transformations matters.

Geometry and TrigonometryNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: the point (3, 5) is reflected in the y-axis. Where is its image?
    Answer(−3, 5). The x-coordinate changes sign.
  2. From Year 9: the point (4, 1) is reflected in the line x = 6. Where is its image?
    Answer(8, 1). The point is 2 to the left of the mirror, so its image is 2 to the right.
  3. From Year 9: the point (5, 1) is translated by the vector with top −2 and bottom 3. Where is its image?
    Answer(3, 4), because 5 − 2 = 3 and 1 + 3 = 4.
Words used in this lesson
isometry
a transformation that keeps all lengths and angles the same. For example: a slide, a turn or a flip.
translation
a slide of every point by the same vector. For example: a tile moved along a wall.
rotation
a turn about a point through an angle. For example: a clock hand moving.
glide reflection
a reflection followed by a slide along the mirror. For example: footprints in sand.
commute
give the same result when the order is swapped. For example: 3 × 5 and 5 × 3 give the same; putting on socks then shoes does not.

The vase between two mirrors

In a hall of mirrors in Isfahan, a tiled vase stands between two parallel mirrors on opposite walls. Its reflection appears in the first wall, and that reflection is reflected again in the second wall. Each mirror is only a reflection, yet the second image appears to have been carried across the hall without being flipped.

Predict first: how far from the vase is the second image, compared with the gap between the mirrors? Use the first figure to test your answer, and the second to see what happens when the mirrors are not parallel but meet at a point.

0
0
(−1, 1)A″, the final image of A (−1, 1)
no movethe single transformation

Reflect in x = 0, then in x = 0: A (−1, 1) goes to (1, 1), then to (−1, 1). Together this is no move.

Show me the steps
  1. Set p = −2 and q = 1: the gap is 3, so the slide is 6 to the right, and A (−1, 1) ends at (5, 1).
  2. Keep p = −2 and q = 1 and press “q first, then p”: the slide is now 6 to the left, and A ends at (−7, 1). Swapping the mirrors reverses the slide.
  3. Set p = 2 and q = −1: the second mirror is to the left of the first, so the slide is 6 to the left and A ends at (−7, 1), the same as the swapped order before.
0
0
no turnthe single transformation
20°angle of P″ from the x-axis

First mirror at 0°, then at 0°: P (at 20°) goes to angle 340°, then to 20°. Together this is no turn.

Show me the steps
  1. Set the first mirror to 15° and the second to 75°: the angle between them is 60°, so the turn is 120° anticlockwise, and P moves from 20° to 140°.
  2. Keep both mirrors and press “second, then first”: the turn is 120° clockwise, and P ends at 260°. The same mirrors in the other order give the opposite turn.
  3. Set both mirrors to 45°: the same mirror used twice gives no turn, and P is back at 20°.

Parallel mirrors. Reflect in x = p and then in x = q. A point at x goes to 2p − x and then to 2q − (2p − x) = x + 2(q − p). The result is a translation of 2(q − p) perpendicular to the mirrors: twice the gap, in the direction from the first mirror to the second.

Mirrors that meet. Two mirrors through a point O at an angle θ to each other give a rotation about O through 2θ, in the direction from the first mirror to the second.

Three mirrors and order. Every isometry is the combination of at most three reflections; a glide reflection needs three. The order of two reflections matters, because swapping them reverses the slide or turn.

Replacing. Any slide or turn can be written as two reflections, in many ways: only the gap or the angle is fixed, not the position of the mirrors.

The slide is twice the gap, not the gap. Mirrors 5 apart give a slide of 10, and mirrors at 20° to each other give a turn of 40°, not 20°. Two reflections never make one reflection: parallel mirrors give a slide, meeting mirrors give a turn. A turn followed by a slide is generally not the same as the slide followed by the turn.

Worked examples

1. Reflect (3, 4) in x = −2 and then in x = 1. Find the image and the single transformation.

  1. First mirror: 2(−2) − 3 = −7, so (−7, 4). Second mirror: 2(1) − (−7) = 9, so (9, 4).
  2. The gap is 1 − (−2) = 3, and 2 × 3 = 6 is the slide to the right: 3 + 6 = 9, a translation of 6 to the right.

2. Which two mirrors through the origin give a rotation of 90° anticlockwise?

  1. The turn is 2θ, so the angle between the mirrors is θ = 45°.
  2. For example the first mirror at 25° and the second at 70°, in that order. Any pair 45° apart, first to second anticlockwise, works.

3. Does the order matter for a quarter turn anticlockwise about the origin and a slide of (4, 0), applied to (1, 2)?

  1. Turn then slide: (1, 2) → (−2, 1) → (2, 1).
  2. Slide then turn: (1, 2) → (5, 2) → (−2, 5).
  3. The images are different, so the order matters for a turn and a slide.

Your go. The first step is given. Reflect (5, 2) in x = 1 and then in x = 4. Find the image.

  1. The gap between the mirrors is 4 − 1 = 3.
  2. Now find the slide, 2 × 3, in the direction from x = 1 towards x = 4, and add it to the x-coordinate.
Check the answer

The slide is 6 to the right, so (5, 2) goes to (11, 2). Check by two reflections: 2(1) − 5 = −3, then 2(4) − (−3) = 11.

Check before you go on

1. A shape is reflected in the line x = −1 and then in the line x = 5. Which single transformation does the same job?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. The point (1, 5) is reflected in the line x = 2 and then in the line x = 9. Find the x-coordinate of the final image.

3. Two mirror lines pass through the origin, one at 20° and one at 55° to the x-axis. A shape is reflected in the first and then in the second. Find the angle of the rotation about the origin, in degrees anticlockwise.

Level 2 · Building

4. A rotation of 130° anticlockwise about the origin is made from two reflections. The first mirror is at 15° to the x-axis. Find the angle of the second mirror, in degrees.

5. The point (2, 4) is rotated 90° anticlockwise about the origin and then translated by the vector with top 3 and bottom 1. Find the x-coordinate of its final image.

6. A point is reflected in x = −3 and then in x = 5. Its final image has x-coordinate 30. Find the x-coordinate of the original point.

Level 3 · Stretch

7. A shape is reflected in three mirror lines through the origin, one after another, at 10°, 40° and 85° to the x-axis. The result is a single reflection. Find the angle of its mirror line, in degrees.

8. The point (2, 3) is reflected in the line x = 1 and then rotated 90° anticlockwise about the origin. Find the x-coordinate of its final image.

Something to think about

Reflect a shape in the x-axis and then slide it 5 to the right. Now do it the other way round: slide it 5 to the right, then reflect in the x-axis. Is the final image the same? What if the slide is 5 up instead of 5 to the right?

One way to think about it

A slide along the mirror, to the right, gives the same image in either order: this is a glide reflection. A slide across the mirror, up, does not: reflecting then sliding up 5 leaves the image 5 above where the reflection puts it, but sliding up 5 and then reflecting puts it 5 below. The order matters whenever the slide has a part across the mirror.

Exit ticket

9. Which pair of transformations always gives the same result in either order?