Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Lesson 1. Translate the point (2, −4) by the vector (−3, 5).
Answer
(−1, 1). 2 + (−3) = −1 and −4 + 5 = 1. - From Year 8, Unit 3, integers. What is the opposite of −6?
Answer
6. The opposite has the same size on the other side of zero. - From Year 8, Unit 3, lesson 2, absolute value and distance. How far apart are the numbers −2 and 5 on a number line?
Answer
7. From −2 to 0 is 2, and from 0 to 5 is 5.
Words used in this lesson
- reflection
- a flip over a line so the shape faces the other way. For example: a face in a mirror.
- mirror line
- the line a shape is flipped over; also called the line of reflection. For example: the y-axis, or x = 3.
- image
- the shape after it is flipped; its corners have a dash. For example: A′B′C′.
- perpendicular
- at a right angle (90 degrees). For example: the dashed lines meet the mirror perpendicularly.
- congruent
- exactly the same size and shape. For example: a shape and its reflection.
Mirror, mirror on the tile
A tile maker in Isfahan draws one triangle of a star pattern, with corners A (−2, 1), B (1, 1) and C (−2, 3), and then makes its mirror image across the horizontal axis of her design, the x-axis. The mirror sits on the line y = 0.
Predict first: where does the corner C (−2, 3) land? Then press the buttons to see the other mirror lines for the same triangle.
Reflected in the y-axis, A (−2, 1) lands on A′ (2, 1). Each corner is as far from the mirror as its image.
Show me the steps
- The y-axis: x changes sign and y stays, so A (−2, 1) goes to (2, 1) and C (−2, 3) to (2, 3). B (1, 1) goes to (−1, 1).
- The x-axis: y changes sign and x stays, so A goes to (−2, −1) and C to (−2, −3).
- The line y = x: the coordinates swap, so A goes to (1, −2) and C to (3, −2). B (1, 1) is on the mirror line and does not move.
- The line x = 3: every corner goes the same distance past the line. A is 5 left of it and lands 5 right of it, at (8, 1).
A reflection flips a shape over a mirror line. Each point and its image are the same distance from the line, on opposite sides, and the line joining them crosses the mirror at a right angle.
In the x-axis, (x, y) → (x, −y). In the y-axis, (x, y) → (−x, y). In the line y = x, (x, y) → (y, x).
Lengths and angles are kept, so a reflection is a congruence transformation, but the order of the corners reverses: here ABC runs anticlockwise round the triangle, and A′B′C′ runs clockwise.
Keeping the distance
The coordinate rules are shortcuts for one idea: the same distance on the other side. Slide the mirror and watch the distances.
The mirror is x = 2. A (−2, 1) is 4 from it and A′ (6, 1) is 4 from it on the other side, so the gap from A to A′ is 8.
Show me the steps
- Mirror x = 2: A (−2, 1) is 4 from the mirror and A′ (6, 1) is 4 from it on the other side. The gap from A to A′ is 8.
- Mirror x = 0, the y-axis: A is 2 from the mirror and A′ (2, 1) is 2 from it. The gap is 4.
- Mirror x = −1, through the triangle: B (1, 1) is 2 from it and lands at (−3, 1); C (−2, 3) is 1 from it and lands at (0, 3). The image overlaps the pre-image.
- Horizontal mirror y = 2: A (−2, 1) is 1 below it and lands at (−2, 3), and C (−2, 3) is 1 above it and lands at (−2, 1). A and C swap places.
The sign change belongs to the axes, not to every reflection. In the x-axis only y changes sign, so a corner at (−2, 1) goes to (−2, −1), not (2, −1). For a mirror that is not an axis, such as x = 3, use the distance: the image is the same distance past the line.
Worked examples
Reflect P (3, −5) in the x-axis. The x-axis flips up and down, so the y-coordinate changes sign and x stays: P′ (3, 5).
Reflect the triangle (1, 1), (4, 2), (2, 5) in the y-axis. Change the sign of every x-coordinate: (−1, 1), (−4, 2), (−2, 5).
Reflect Q (5, 1) in the line y = x. Swap the coordinates: Q′ (1, 5). Check: both points are the same distance from the line, and the line joining them slopes down from left to right, at a right angle to the mirror.
Reflect R (−4, 3) in the line x = −1.
- R is 3 to the left of the line: −1 − (−4) = 3.
- The image is 3 to the right of the line: −1 + 3 = 2.
- The y-coordinate does not change, so R′ (2, 3).
Your go. The first step is given. Reflect S (−4, −3) in the x-axis.
- The x-axis flips up and down, so x stays at −4.
- Find the new y-coordinate, then write S′.
Check the answer
y changes sign: −3 becomes 3, so S′ is (−4, 3).
Check before you go on
1. The point (−6, 4) is reflected in the x-axis. Where does it land?
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 (7, −3) is reflected in the y-axis. Its image is (x, y). What is x?
3. The point (5, 8) is reflected in the line y = x. What is the y-coordinate of the image?
Level 2 · Building
4. Q (−3, 5) is reflected in the line x = 2. What is the x-coordinate of the image?
5. A corner of a festival lantern is at (2, 6). The lantern is reflected in the horizontal line y = 4. What is the y-coordinate of the image?
6. The point (2, 3) is reflected onto (8, 3). Which line is the mirror?
Level 3 · Stretch
7. The point (4, −1) is reflected in the x-axis, and the image is then reflected in the y-axis. The final image is (x, y). What is x + y?
8. A point and its image after reflection in the line x = 6 are 10 units apart. How far is the point from the line x = 6?
Something to think about
In the first figure, one corner of the triangle did not move when the mirror was the line y = x. Which corner, and what is special about it? Could a corner stay put in the y-axis mirror?
One way to think about it
B (1, 1) did not move: it lies on the line y = x, so it is zero distance from the mirror and its image is itself. In the y-axis mirror a corner stays put only if it has x = 0, that is, if it sits on the y-axis. Every point on the mirror line is its own image.
Exit ticket
9. Which point is the image of (−3, −2) in the line y = x?