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

What single move is two reflections, and does the order matter?

One transformation followed by another is a new transformation. This lesson asks which single move two reflections make, how that depends on the mirrors, and whether swapping the order can change the result.

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: a point (4, −3) is reflected in the y-axis. What are the coordinates of its image?
    Answer(−4, −3). A reflection in the y-axis changes the sign of x and keeps y.
  2. From the last lesson: the point (5, 1) is translated by the column vector with top −2 and bottom 6. Where is its image?
    Answer(3, 7), because 5 + (−2) = 3 and 1 + 6 = 7.
  3. From Year 8: two straight lines cross. One angle is 70°. What is the angle next to it on a straight line?
    Answer110°, because angles on a straight line add up to 180°.
Words used in this lesson
combined transformation
one transformation followed by another. For example: a turn then a slide.
parallel
never meeting, always the same distance apart. For example: railway lines.
glide reflection
a reflection followed by a slide along the mirror line. For example: footprints in sand.
identity
the transformation that moves nothing. For example: turning 360°.
orientation
which way round a shape is: clockwise or anticlockwise. For example: the order of the corners A, B, C.
commute
give the same result in either order. For example: putting on a hat and a scarf: either order gives the same result.

A border pattern

A weaver in Kyoto prints a repeating border. She stamps a motif, flips it in a mirror along the right edge of the stamp, and then flips that image in a second mirror, one panel further along. The first mirror and the second are parallel.

Predict first: after the two flips, is the motif a mirror image of the original, a slid copy, or a turned copy? Does the distance it moves depend on how far apart the mirrors are? Write down a reason, then use the first figure.

−2
1
20
(5, 1)A″, the final image of A (−1, 1)
slide of 6 to the rightthe single transformation

Reflect in x = −2, then in x = 1: A (−1, 1) goes to (−3, 1), then to (5, 1). Together this is a slide of 6 to the right.

Show me the steps
  1. Parallel mirrors x = −2 then x = 1: the gap is 3, and the triangle slides 2 × 3 = 6 to the right. A (−1, 1) ends at (5, 1).
  2. Make a = 0 and b = 1: the gap is 1, so the slide is 2. A ends at (1, 1). Doubling the gap doubles the slide.
  3. Make a = 2 and b = −1: the second mirror is to the left of the first, so the slide is 6 to the left. A ends at (−7, 1).
  4. Press the meeting mirrors with an angle of 40°: the x-axis and then a line at 40° make a rotation of 80° anticlockwise about the origin. A goes to about (−1.16, −0.81).

Two reflections make a translation or a rotation. In parallel mirrors, the result is a translation of twice the gap between them, perpendicular to the mirrors. In mirrors that meet, the result is a rotation about the meeting point through twice the angle between them.

Every isometry is a reflection, a translation, a rotation or a glide reflection (a reflection with a slide along the mirror), and can be built from at most three reflections (you will not be asked for this today). Order matters: a turn then a slide is usually a different transformation from a slide then a turn.

2
1
(3, 0)A after turn, then slide
(2, −3)A after slide, then turn

Slide top 2, bottom 1. Turn then slide: A (1, 1) goes to (3, 0) and C (1, 2) to (4, 0). Slide then turn: A goes to (2, −3) and C to (3, −3). The images are different.

Show me the steps
  1. Slide top 2, bottom 1: turn then slide puts A at (3, 0); slide then turn puts A at (2, −3). The two final triangles are in different places.
  2. Make p = 0 and q = 0: there is no slide, so both orders give the same turn. A goes to (1, −1) either way.
  3. Make p = 3 and q = 3: turn then slide puts A at (4, 2); slide then turn puts A at (4, −4). The x-coordinates agree but the y-coordinates do not.
  4. Make p = −2 and q = 0: turn then slide puts A at (−1, −1); slide then turn puts A at (1, 1), which is back where it started.

Name the order. “A then B” means A first. In a description, say which mirror comes first, because the direction of the translation or the turn depends on it. Reflecting in x = 4 and then x = 1 slides left, not right.

Worked examples

1. A shape is reflected in x = −1 and then in x = 4. Describe the single transformation.

  1. The mirrors are parallel, and the gap is 4 − (−1) = 5.
  2. The shape slides 2 × 5 = 10 units to the right, because the second mirror is to the right of the first.

2. Two mirror lines meet at the origin at an angle of 35°. Describe the single transformation.

  1. Two reflections in lines that meet make a rotation about the meeting point.
  2. The angle is twice 35°, so it is a rotation of 70° about the origin, from the first mirror towards the second.

3. The point (1, 3) is reflected in the x-axis and then translated by the column vector with top 2 and bottom 1. Then do it the other way round.

  1. Reflect first: (1, 3) goes to (1, −3); then translate: (3, −2).
  2. Translate first: (1, 3) goes to (3, 4); then reflect in the x-axis: (3, −4). The order gave two different points.

Your go. The first step is given. The point (1, 4) is reflected in x = 2 and then in x = 5. Where does it end?

  1. The mirrors are parallel with a gap of 3, so the point slides 2 × 3 = 6 to the right.
  2. Add 6 to the x-coordinate. The y-coordinate does not change.
Check the answer

The end point is (7, 4). By hand: reflect in x = 2 to (3, 4), then in x = 5 to (7, 4).

Check before you go on

1. A shape is reflected in the line x = 2 and then the image is reflected in the line x = 9. 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. A shape is reflected in the line x = 1 and then in the line x = 5. How far does it end up from where it started, in units?

3. The point (3, 4) is reflected in the x-axis and then in the y-axis. What is the x-coordinate of the final image?

Level 2 · Building

4. Two mirror lines meet at a point at an angle of 25°. A shape is reflected in one and then in the other. Through what angle, in degrees, is it rotated about that point?

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

6. Now do it the other way round. The point (4, 2) is translated by the column vector with top 3 and bottom −1, then rotated 90° clockwise about the origin. What is the y-coordinate of the final image?

Level 3 · Stretch

7. The point (5, 4) is rotated 180° about the centre (2, 1). What is the x-coordinate of the image?

8. A shape is rotated through 130° about the origin by two reflections in lines through the origin. What is the angle between the two mirror lines, in degrees?

Something to think about

Two reflections in parallel mirrors give a translation. What do three reflections in three parallel mirrors give? Try x = 0, then x = 1, then x = 3 on the point (2, 5).

One way to think about it

A single reflection again. The point (2, 5) goes to (−2, 5), then to (4, 5), then back to (2, 5). It did not move, because it lies on the mirror x = 2, which is the single reflection that does the same job as the three (3 − 1 + 0 = 2). An odd number of reflections reverses the orientation, so three reflections can never make a translation or a rotation.

Exit ticket

9. Which pair of transformations always gives the same image, whichever one is done first?