GCSE Maths: Reflection (Higher)
Part of the GCSE Maths transformations and similarity chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Reflection
A transformation changes a shape in some way. The shape we start with is called the object and the new shape is called the image. In this chapter you will meet four transformations: reflection, rotation, translation and enlargement. The first three only move the shape, so the image is exactly the same size and shape as the object. These shapes are called congruent. Enlargement changes the size, so the image is similar to the object but not congruent.
A reflection gives a mirror image. Imagine standing a mirror on the mirror line and looking at the shape in it. Each point of the image is the same distance from the mirror line as the matching point of the object, but on the opposite side. The line joining a point to its image crosses the mirror line at a right angle. That is why the mirror line is exactly half-way between a point and its image.
On a coordinate grid the mirror line is usually one of four kinds of line:
- A vertical line x = a number, such as x = 4. The y-axis is the line x = 0.
- A horizontal line y = a number, such as y = −1. The x-axis is the line y = 0.
- The diagonal y = x, which goes through (1, 1), (2, 2), (3, 3) and so on.
- The other diagonal y = −x, which goes through (1, −1), (2, −2), (−1, 1) and so on.
For a vertical or horizontal mirror line, count the squares from each vertex to the line, then count the same number of squares on the other side. For the diagonals there is a shortcut, shown in the picture: a reflection in y = x swaps the coordinates, (a, b) → (b, a), and a reflection in y = −x swaps them and changes both signs, (a, b) → (−b, −a). The axes themselves are easy as well: reflecting in the x-axis changes the sign of y, and reflecting in the y-axis changes the sign of x.
Describing a reflection. To describe a reflection fully you must say that it is a reflection and give the equation of the mirror line. To find the line, look at a point and its image: the line is exactly half-way between them. If you have tracing paper, you can also trace the object, flip the tracing paper over and slide it until it lands on the image, then draw along the fold.
Key ideas
- A reflection flips a shape over a mirror line. The image is the same size and shape (congruent) but a mirror image.
- Each point and its image are the same distance from the mirror line, on opposite sides.
- Vertical line x = a: the y-coordinate stays the same. Horizontal line y = b: the x-coordinate stays the same.
- In y = x: (a, b) → (b, a). In y = −x: (a, b) → (−b, −a).
- To describe a reflection, give the equation of the mirror line.
- A point on the mirror line is invariant: it does not move.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. To reflect a shape, reflect every vertex over the mirror line x = 5, then join the image points in the same order.
- Draw the mirror line. The line x = 5 is a vertical line: every point on it has x = 5. Everything in the picture will flip over this line.Draw a vertical line through x = 5.
- Reflect each vertex of A in turn. For a reflection in a vertical line x = 5, the y-coordinate stays the same. Count how far each point is from the line (sideways), then go the same distance on the other side.(1, 1): 4 left of the line, so the image is 4 right of the line. x = 5 + 4 = 9. Image (9, 1)
(4, 1): 1 left of the line, so the image is 1 right of the line. x = 5 + 1 = 6. Image (6, 1)
(1, 3): 4 left of the line, so the image is 4 right of the line. x = 5 + 4 = 9. Image (9, 3) - Plot the image points and join them up in the same order to make shape B. The image has the same size and shape as the object but is the other way round, like a mirror image.B has vertices at (9, 1), (6, 1), (9, 3).
Check. The mirror line should be exactly half-way between a point and its image. The midpoint of (1, 1) and (9, 1) is (1 + 9) ÷ 2, (1 + 1) ÷ 2 = (5, 1), which lies on the line x = 5. ✓
Watch out. Measure the distance from the mirror line, not from the y-axis. The line is x = 5, so distances start from there.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. To reflect a shape, reflect every vertex over the mirror line y = −1, then join the image points in the same order.
- Draw the mirror line. The line y = −1 is a horizontal line: every point on it has y = −1. Everything in the picture will flip over this line.Draw a horizontal line through y = −1.
- Reflect each vertex of A in turn. For a reflection in a horizontal line y = −1, the x-coordinate stays the same. Count how far each point is from the line (up or down), then go the same distance on the other side.(1, 2): 3 above the line, so the image is 3 below the line. y = −1 − 3 = −4. Image (1, −4)
(3, 2): 3 above the line, so the image is 3 below the line. y = −1 − 3 = −4. Image (3, −4)
(3, 3): 4 above the line, so the image is 4 below the line. y = −1 − 4 = −5. Image (3, −5)
(2, 3): 4 above the line, so the image is 4 below the line. y = −1 − 4 = −5. Image (2, −5)
(2, 5): 6 above the line, so the image is 6 below the line. y = −1 − 6 = −7. Image (2, −7)
(1, 5): 6 above the line, so the image is 6 below the line. y = −1 − 6 = −7. Image (1, −7) - Plot the image points and join them up in the same order to make shape B. The image has the same size and shape as the object but is the other way round, like a mirror image.B has vertices at (1, −4), (3, −4), (3, −5), (2, −5), (2, −7), (1, −7).
Check. The mirror line should be exactly half-way between a point and its image. The midpoint of (1, 2) and (1, −4) is (1 + 1) ÷ 2, (2 + −4) ÷ 2 = (1, −1), which lies on the line y = −1. ✓
Watch out. Measure the distance from the mirror line, not from the x-axis. The line is y = −1, so distances start from there.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. To reflect a shape, reflect every vertex over the mirror line y = x, then join the image points in the same order.
- Draw the mirror line. The line y = x is a diagonal line. Every point on it has the same x and y coordinates (for example (0, 0), (1, 1), (2, 2), (3, 3)).Draw a straight line through (0, 0), (1, 1), (2, 2) and (3, 3).
- Reflect each vertex of A in turn. A mirror along y = x swaps the two coordinates: the point (a, b) becomes (b, a). This works because the diagonal is the line where the x and y coordinates are equal.(3, 1) → (1, 3)
(5, 1) → (1, 5)
(6, 3) → (3, 6)
(3, 2) → (2, 3) - Plot the image points and join them up in the same order to make shape B. The image has the same size and shape as the object but is the other way round, like a mirror image.B has vertices at (1, 3), (1, 5), (3, 6), (2, 3).
Check. The mirror line should be exactly half-way between a point and its image. The midpoint of (3, 1) and (1, 3) is (3 + 1) ÷ 2, (1 + 3) ÷ 2 = (2, 2), which lies on the line y = x. ✓
Watch out. Reflecting in y = x swaps the coordinates; it does not change any signs. (Changing signs is for y = −x.)
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. To reflect a shape, reflect every vertex over the mirror line y = −x, then join the image points in the same order.
- Draw the mirror line. The line y = −x is a diagonal line sloping down to the right. Every point on it has y equal to the opposite of x (for example (−2, 2), (−1, 1), (0, 0), (1, −1), (2, −2)).Draw a straight line through (−2, 2), (0, 0) and (2, −2).
- Reflect each vertex of A in turn. A mirror along y = −x swaps the coordinates and changes both signs: the point (a, b) becomes (−b, −a). For example (2, 1) goes to (−1, −2).(−4, 1) → (−1, 4)
(−1, 2) → (−2, 1)
(−3, 4) → (−4, 3) - Plot the image points and join them up in the same order to make shape B. The image has the same size and shape as the object but is the other way round, like a mirror image.B has vertices at (−1, 4), (−2, 1), (−4, 3).
Check. The mirror line should be exactly half-way between a point and its image. The midpoint of (−4, 1) and (−1, 4) is (−4 + −1) ÷ 2, (1 + 4) ÷ 2 = (−2.5, 2.5), which lies on the line y = −x. ✓
Watch out. Reflecting in y = −x swaps the coordinates and changes both signs. Swapping alone gives a reflection in y = x.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. A reflection flips the shape over. All other transformations of this type keep the shape facing the same way.Compare a side: it is 3 long in A and 3 long in B.
The size is unchanged, but B is the mirror image of A: if you go round the corners in the same order, one shape is clockwise and the other is anticlockwise. So the transformation is a reflection. - Pick vertices and their images, and find the midpoints. A mirror line is exactly half-way between an object point and its image, so find the midpoint of a point and its image.(1, 1) and (11, 1): midpoint = ((1 + 11) ÷ 2, (1 + 1) ÷ 2) = (6, 1)
(4, 1) and (8, 1): midpoint = ((4 + 8) ÷ 2, (1 + 1) ÷ 2) = (6, 1) - Write the equation of the mirror line. The midpoints all have x = 6. The points are in a vertical line, so the mirror line is the vertical line x = 6.Mirror line: x = 6
- Write the full description. We have found every piece of information needed for this transformation.Reflection in the line x = 6
Check. Reflect (4, 1) in x = 6 and we get (8, 1), which matches (8, 1) in B. ✓
Watch out. A reflection needs the equation of the mirror line. Writing just “reflection” or “reflect in the x-axis” for a line like x = 3 is wrong.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. A reflection flips the shape over. All other transformations of this type keep the shape facing the same way.Compare the areas: 4 square units for A and 4 for B.
The size is unchanged, but B is the mirror image of A: if you go round the corners in the same order, one shape is clockwise and the other is anticlockwise. So the transformation is a reflection. - Compare the coordinates of a vertex and its image. For the diagonal mirror lines y = x and y = −x the coordinates change in a special way.The coordinates have been swapped: (2, 1) became (1, 2).
- Use the midpoint as a check. The mirror line must pass through the midpoint of a point and its image.Midpoint of (2, 1) and (1, 2) = (1.5, 1.5), which lies on y = x.
- Write the equation of the mirror line. A reflection that swaps x and y is a reflection in the line y = x.Mirror line: y = x
- Write the full description. We have found every piece of information needed for this transformation.Reflection in the line y = x
Check. Reflect (5, 2) in y = x and we get (2, 5), which matches (2, 5) in B. ✓
Watch out. A reflection needs the equation of the mirror line. Writing just “reflection” or “reflect in the x-axis” for a line like x = 3 is wrong.
Watch out
Always measure from the mirror line, not from the axes (unless the axis is the mirror line). For a line like x = 4, a point at x = 1 is 3 squares from the line, so its image is at x = 4 + 3 = 7. Also, in y = −x do not forget to change both signs as well as swapping.Exercise 21A Reflection
Non-calculatorKey words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. To reflect a shape, reflect every vertex over the mirror line x = 0, then join the image points in the same order.
- Draw the mirror line. The line x = 0 is a vertical line: every point on it has x = 0. Everything in the picture will flip over this line.Draw a vertical line through x = 0.
- Reflect each vertex of A in turn. For a reflection in a vertical line x = 0, the y-coordinate stays the same. Count how far each point is from the line (sideways), then go the same distance on the other side.(2, 2): 2 right of the line, so the image is 2 left of the line. x = 0 − 2 = −2. Image (−2, 2)
(5, 2): 5 right of the line, so the image is 5 left of the line. x = 0 − 5 = −5. Image (−5, 2)
(2, 4): 2 right of the line, so the image is 2 left of the line. x = 0 − 2 = −2. Image (−2, 4) - Plot the image points and join them up in the same order to make shape B. The image has the same size and shape as the object but is the other way round, like a mirror image.B has vertices at (−2, 2), (−5, 2), (−2, 4).
Check. The mirror line should be exactly half-way between a point and its image. The midpoint of (2, 2) and (−2, 2) is (2 + −2) ÷ 2, (2 + 2) ÷ 2 = (0, 2), which lies on the line x = 0. ✓
Watch out. Measure the distance from the mirror line, not from the y-axis. The line is x = 0, so distances start from there.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. To reflect a shape, reflect every vertex over the mirror line y = 0, then join the image points in the same order.
- Draw the mirror line. The line y = 0 is a horizontal line: every point on it has y = 0. Everything in the picture will flip over this line.Draw a horizontal line through y = 0.
- Reflect each vertex of A in turn. For a reflection in a horizontal line y = 0, the x-coordinate stays the same. Count how far each point is from the line (up or down), then go the same distance on the other side.(2, 2): 2 above the line, so the image is 2 below the line. y = 0 − 2 = −2. Image (2, −2)
(5, 2): 2 above the line, so the image is 2 below the line. y = 0 − 2 = −2. Image (5, −2)
(2, 4): 4 above the line, so the image is 4 below the line. y = 0 − 4 = −4. Image (2, −4) - Plot the image points and join them up in the same order to make shape B. The image has the same size and shape as the object but is the other way round, like a mirror image.B has vertices at (2, −2), (5, −2), (2, −4).
Check. The mirror line should be exactly half-way between a point and its image. The midpoint of (2, 2) and (2, −2) is (2 + 2) ÷ 2, (2 + −2) ÷ 2 = (2, 0), which lies on the line y = 0. ✓
Watch out. Measure the distance from the mirror line, not from the x-axis. The line is y = 0, so distances start from there.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. To reflect a shape, reflect every vertex over the mirror line x = 6, then join the image points in the same order.
- Draw the mirror line. The line x = 6 is a vertical line: every point on it has x = 6. Everything in the picture will flip over this line.Draw a vertical line through x = 6.
- Reflect each vertex of A in turn. For a reflection in a vertical line x = 6, the y-coordinate stays the same. Count how far each point is from the line (sideways), then go the same distance on the other side.(2, 2): 4 left of the line, so the image is 4 right of the line. x = 6 + 4 = 10. Image (10, 2)
(5, 2): 1 left of the line, so the image is 1 right of the line. x = 6 + 1 = 7. Image (7, 2)
(2, 4): 4 left of the line, so the image is 4 right of the line. x = 6 + 4 = 10. Image (10, 4) - Plot the image points and join them up in the same order to make shape B. The image has the same size and shape as the object but is the other way round, like a mirror image.B has vertices at (10, 2), (7, 2), (10, 4).
Check. The mirror line should be exactly half-way between a point and its image. The midpoint of (2, 2) and (10, 2) is (2 + 10) ÷ 2, (2 + 2) ÷ 2 = (6, 2), which lies on the line x = 6. ✓
Watch out. Measure the distance from the mirror line, not from the y-axis. The line is x = 6, so distances start from there.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. To reflect a shape, reflect every vertex over the mirror line y = 6, then join the image points in the same order.
- Draw the mirror line. The line y = 6 is a horizontal line: every point on it has y = 6. Everything in the picture will flip over this line.Draw a horizontal line through y = 6.
- Reflect each vertex of A in turn. For a reflection in a horizontal line y = 6, the x-coordinate stays the same. Count how far each point is from the line (up or down), then go the same distance on the other side.(2, 2): 4 below the line, so the image is 4 above the line. y = 6 + 4 = 10. Image (2, 10)
(5, 2): 4 below the line, so the image is 4 above the line. y = 6 + 4 = 10. Image (5, 10)
(2, 4): 2 below the line, so the image is 2 above the line. y = 6 + 2 = 8. Image (2, 8) - Plot the image points and join them up in the same order to make shape B. The image has the same size and shape as the object but is the other way round, like a mirror image.B has vertices at (2, 10), (5, 10), (2, 8).
Check. The mirror line should be exactly half-way between a point and its image. The midpoint of (2, 2) and (2, 10) is (2 + 2) ÷ 2, (2 + 10) ÷ 2 = (2, 6), which lies on the line y = 6. ✓
Watch out. Measure the distance from the mirror line, not from the x-axis. The line is y = 6, so distances start from there.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. To reflect a shape, reflect every vertex over the mirror line y = −2, then join the image points in the same order.
- Draw the mirror line. The line y = −2 is a horizontal line: every point on it has y = −2. Everything in the picture will flip over this line.Draw a horizontal line through y = −2.
- Reflect each vertex of A in turn. For a reflection in a horizontal line y = −2, the x-coordinate stays the same. Count how far each point is from the line (up or down), then go the same distance on the other side.(1, 2): 4 above the line, so the image is 4 below the line. y = −2 − 4 = −6. Image (1, −6)
(4, 2): 4 above the line, so the image is 4 below the line. y = −2 − 4 = −6. Image (4, −6)
(3, 4): 6 above the line, so the image is 6 below the line. y = −2 − 6 = −8. Image (3, −8)
(1, 4): 6 above the line, so the image is 6 below the line. y = −2 − 6 = −8. Image (1, −8) - Plot the image points and join them up in the same order to make shape B. The image has the same size and shape as the object but is the other way round, like a mirror image.B has vertices at (1, −6), (4, −6), (3, −8), (1, −8).
Check. The mirror line should be exactly half-way between a point and its image. The midpoint of (1, 2) and (1, −6) is (1 + 1) ÷ 2, (2 + −6) ÷ 2 = (1, −2), which lies on the line y = −2. ✓
Watch out. Measure the distance from the mirror line, not from the x-axis. The line is y = −2, so distances start from there.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. To reflect a shape, reflect every vertex over the mirror line x = −1, then join the image points in the same order.
- Draw the mirror line. The line x = −1 is a vertical line: every point on it has x = −1. Everything in the picture will flip over this line.Draw a vertical line through x = −1.
- Reflect each vertex of A in turn. For a reflection in a vertical line x = −1, the y-coordinate stays the same. Count how far each point is from the line (sideways), then go the same distance on the other side.(2, 1): 3 right of the line, so the image is 3 left of the line. x = −1 − 3 = −4. Image (−4, 1)
(4, 1): 5 right of the line, so the image is 5 left of the line. x = −1 − 5 = −6. Image (−6, 1)
(4, 2): 5 right of the line, so the image is 5 left of the line. x = −1 − 5 = −6. Image (−6, 2)
(3, 2): 4 right of the line, so the image is 4 left of the line. x = −1 − 4 = −5. Image (−5, 2)
(3, 4): 4 right of the line, so the image is 4 left of the line. x = −1 − 4 = −5. Image (−5, 4)
(2, 4): 3 right of the line, so the image is 3 left of the line. x = −1 − 3 = −4. Image (−4, 4) - Plot the image points and join them up in the same order to make shape B. The image has the same size and shape as the object but is the other way round, like a mirror image.B has vertices at (−4, 1), (−6, 1), (−6, 2), (−5, 2), (−5, 4), (−4, 4).
Check. The mirror line should be exactly half-way between a point and its image. The midpoint of (2, 1) and (−4, 1) is (2 + −4) ÷ 2, (1 + 1) ÷ 2 = (−1, 1), which lies on the line x = −1. ✓
Watch out. Measure the distance from the mirror line, not from the y-axis. The line is x = −1, so distances start from there.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. To reflect a shape, reflect every vertex over the mirror line y = x, then join the image points in the same order.
- Draw the mirror line. The line y = x is a diagonal line. Every point on it has the same x and y coordinates (for example (0, 0), (1, 1), (2, 2), (3, 3)).Draw a straight line through (0, 0), (1, 1), (2, 2) and (3, 3).
- Reflect each vertex of A in turn. A mirror along y = x swaps the two coordinates: the point (a, b) becomes (b, a). This works because the diagonal is the line where the x and y coordinates are equal.(2, 1) → (1, 2)
(5, 1) → (1, 5)
(2, 3) → (3, 2) - Plot the image points and join them up in the same order to make shape B. The image has the same size and shape as the object but is the other way round, like a mirror image.B has vertices at (1, 2), (1, 5), (3, 2).
Check. The mirror line should be exactly half-way between a point and its image. The midpoint of (2, 1) and (1, 2) is (2 + 1) ÷ 2, (1 + 2) ÷ 2 = (1.5, 1.5), which lies on the line y = x. ✓
Watch out. Reflecting in y = x swaps the coordinates; it does not change any signs. (Changing signs is for y = −x.)
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. To reflect a shape, reflect every vertex over the mirror line y = −x, then join the image points in the same order.
- Draw the mirror line. The line y = −x is a diagonal line sloping down to the right. Every point on it has y equal to the opposite of x (for example (−2, 2), (−1, 1), (0, 0), (1, −1), (2, −2)).Draw a straight line through (−2, 2), (0, 0) and (2, −2).
- Reflect each vertex of A in turn. A mirror along y = −x swaps the coordinates and changes both signs: the point (a, b) becomes (−b, −a). For example (2, 1) goes to (−1, −2).(2, 1) → (−1, −2)
(4, 1) → (−1, −4)
(5, 3) → (−3, −5)
(2, 2) → (−2, −2) - Plot the image points and join them up in the same order to make shape B. The image has the same size and shape as the object but is the other way round, like a mirror image.B has vertices at (−1, −2), (−1, −4), (−3, −5), (−2, −2).
Check. The mirror line should be exactly half-way between a point and its image. The midpoint of (2, 1) and (−1, −2) is (2 + −1) ÷ 2, (1 + −2) ÷ 2 = (0.5, −0.5), which lies on the line y = −x. ✓
Watch out. Reflecting in y = −x swaps the coordinates and changes both signs. Swapping alone gives a reflection in y = x.
Key words. A transformation changes the position or size of a shape. The original shape is the object; the new shape is the image.
Method. Apply the transformation to the coordinates of P.
- Remember what this transformation does. Reflecting in a horizontal line y = 0: the x-coordinate stays the same and the new y is 2 × 0 − y.Reflection in the line y = 0
- Apply it to the point. Follow the rule using the coordinates of the point.(3, −2) → (3, 2 × 0 − (−2)) = (3, 2)
- Write the coordinates of the image. The image of P is called P'.P' = (3, 2)
Check. Look at the picture: P and P' are placed as the transformation requires. ✓
Watch out. Take care with minus signs: write the new coordinates carefully, one at a time.
Key words. A transformation changes the position or size of a shape. The original shape is the object; the new shape is the image.
Method. Apply the transformation to the coordinates of P.
- Remember what this transformation does. Reflecting in a vertical line x = 0: the y-coordinate stays the same and the new x is 2 × 0 − x (the same distance on the other side of the line).Reflection in the line x = 0
- Apply it to the point. Follow the rule using the coordinates of the point.(3, −2) → (2 × 0 − 3, −2) = (−3, −2)
- Write the coordinates of the image. The image of P is called P'.P' = (−3, −2)
Check. Look at the picture: P and P' are placed as the transformation requires. ✓
Watch out. Take care with minus signs: write the new coordinates carefully, one at a time.
Key words. A transformation changes the position or size of a shape. The original shape is the object; the new shape is the image.
Method. Apply the transformation to the coordinates of P.
- Remember what this transformation does. Reflecting in y = x swaps the coordinates: (a, b) → (b, a).Reflection in the line y = x
- Apply it to the point. Follow the rule using the coordinates of the point.(3, −2) → (−2, 3)
- Write the coordinates of the image. The image of P is called P'.P' = (−2, 3)
Check. Look at the picture: P and P' are placed as the transformation requires. ✓
Watch out. Take care with minus signs: write the new coordinates carefully, one at a time.
Key words. A transformation changes the position or size of a shape. The original shape is the object; the new shape is the image.
Method. Apply the transformation to the coordinates of P.
- Remember what this transformation does. Reflecting in y = −x swaps the coordinates and changes both signs: (a, b) → (−b, −a).Reflection in the line y = −x
- Apply it to the point. Follow the rule using the coordinates of the point.(3, −2) → (2, −3)
- Write the coordinates of the image. The image of P is called P'.P' = (2, −3)
Check. Look at the picture: P and P' are placed as the transformation requires. ✓
Watch out. Take care with minus signs: write the new coordinates carefully, one at a time.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. To reflect a shape, reflect every vertex over the mirror line x = 3, then join the image points in the same order.
- Draw the mirror line. The line x = 3 is a vertical line: every point on it has x = 3. Everything in the picture will flip over this line.Draw a vertical line through x = 3.
- Reflect each vertex of A in turn. For a reflection in a vertical line x = 3, the y-coordinate stays the same. Count how far each point is from the line (sideways), then go the same distance on the other side.(1, 1): 2 left of the line, so the image is 2 right of the line. x = 3 + 2 = 5. Image (5, 1)
(4, 2): 1 right of the line, so the image is 1 left of the line. x = 3 − 1 = 2. Image (2, 2)
(2, 4): 1 left of the line, so the image is 1 right of the line. x = 3 + 1 = 4. Image (4, 4) - Plot the image points and join them up in the same order to make shape B. The image has the same size and shape as the object but is the other way round, like a mirror image.B has vertices at (5, 1), (2, 2), (4, 4).
Check. The mirror line should be exactly half-way between a point and its image. The midpoint of (1, 1) and (5, 1) is (1 + 5) ÷ 2, (1 + 1) ÷ 2 = (3, 1), which lies on the line x = 3. ✓
Watch out. Measure the distance from the mirror line, not from the y-axis. The line is x = 3, so distances start from there.
Method. Check what a reflection in y = x does to the coordinates.
- Recall the rule. A reflection in y = x swaps the two coordinates: (a, b) → (b, a). No signs change.(2, 1) → (1, 2)
- See what Priya did. She swapped the coordinates and changed both signs. That is the rule for a reflection in y = −x.Priya: (2, 1) → (−1, −2)
- Write the correct image. Swap the coordinates of every vertex and change nothing else.(1, 2), (1, 5), (3, 2)
Check. The point (2, 1) is 1 square below the line y = x; its image (1, 2) is 1 square above it. ✓
Watch out. Both diagonals swap the coordinates. Only y = −x also changes the signs.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. A reflection flips the shape over. All other transformations of this type keep the shape facing the same way.Compare a side: it is 3 long in A and 3 long in B.
The size is unchanged, but B is the mirror image of A: if you go round the corners in the same order, one shape is clockwise and the other is anticlockwise. So the transformation is a reflection. - Pick vertices and their images, and find the midpoints. A mirror line is exactly half-way between an object point and its image, so find the midpoint of a point and its image.(1, 1) and (11, 1): midpoint = ((1 + 11) ÷ 2, (1 + 1) ÷ 2) = (6, 1)
(4, 1) and (8, 1): midpoint = ((4 + 8) ÷ 2, (1 + 1) ÷ 2) = (6, 1) - Write the equation of the mirror line. The midpoints all have x = 6. The points are in a vertical line, so the mirror line is the vertical line x = 6.Mirror line: x = 6
- Write the full description. We have found every piece of information needed for this transformation.Reflection in the line x = 6
Check. Reflect (4, 1) in x = 6 and we get (8, 1), which matches (8, 1) in B. ✓
Watch out. A reflection needs the equation of the mirror line. Writing just “reflection” or “reflect in the x-axis” for a line like x = 3 is wrong.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. A reflection flips the shape over. All other transformations of this type keep the shape facing the same way.Compare a side: it is 3 long in A and 3 long in B.
The size is unchanged, but B is the mirror image of A: if you go round the corners in the same order, one shape is clockwise and the other is anticlockwise. So the transformation is a reflection. - Pick vertices and their images, and find the midpoints. A mirror line is exactly half-way between an object point and its image, so find the midpoint of a point and its image.(1, 2) and (1, −4): midpoint = ((1 + 1) ÷ 2, (2 + −4) ÷ 2) = (1, −1)
(4, 2) and (4, −4): midpoint = ((4 + 4) ÷ 2, (2 + −4) ÷ 2) = (4, −1) - Write the equation of the mirror line. The midpoints all have y = −1. The points are in a horizontal line, so the mirror line is the horizontal line y = −1.Mirror line: y = −1
- Write the full description. We have found every piece of information needed for this transformation.Reflection in the line y = −1
Check. Reflect (4, 2) in y = −1 and we get (4, −4), which matches (4, −4) in B. ✓
Watch out. A reflection needs the equation of the mirror line. Writing just “reflection” or “reflect in the x-axis” for a line like x = 3 is wrong.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. A reflection flips the shape over. All other transformations of this type keep the shape facing the same way.Compare the areas: 4 square units for A and 4 for B.
The size is unchanged, but B is the mirror image of A: if you go round the corners in the same order, one shape is clockwise and the other is anticlockwise. So the transformation is a reflection. - Compare the coordinates of a vertex and its image. For the diagonal mirror lines y = x and y = −x the coordinates change in a special way.The coordinates have been swapped and both signs changed: (3, 1) became (−1, −3).
- Use the midpoint as a check. The mirror line must pass through the midpoint of a point and its image.Midpoint of (3, 1) and (−1, −3) = (1, −1), which lies on y = −x.
- Write the equation of the mirror line. A reflection that swaps x and y and changes both signs is a reflection in the line y = −x.Mirror line: y = −x
- Write the full description. We have found every piece of information needed for this transformation.Reflection in the line y = −x
Check. Reflect (6, 2) in y = −x and we get (−2, −6), which matches (−2, −6) in B. ✓
Watch out. A reflection needs the equation of the mirror line. Writing just “reflection” or “reflect in the x-axis” for a line like x = 3 is wrong.
Method. The mirror line is exactly half-way between a point and its image.
- Work out the x-coordinate half-way between 4 and 10. Add the two x-coordinates and halve (the midpoint).(4 + 10) ÷ 2 = 14 ÷ 2 = 7
- Write the equation. The line is vertical (the y-coordinates are equal, so the point moved sideways), so its equation is x = a number.x = 7
Check. P is 7 − 4 = 3 to the left of the line and P′ is 10 − 7 = 3 to the right. ✓
Watch out. Do not say x = 3 or x = 6. Find the midpoint, which is half-way between the two x-values.
Method. Think about what happens to one point, then use it for the whole shape.
- Follow one point. Take the point (1, 2) on A. It is 2 to the left of the line.Reflect in x = 3: (1, 2) → (5, 2) (2 to the right of the line)
- Reflect again. The point (5, 2) is 2 to the right of the line, so its image is 2 to the left of the line.(5, 2) → (1, 2)
- Conclude. The point came back to where it started. The same happens to every point, so the whole shape is back where it started.C is in the same place as A.
Check. A reflection is its own inverse: doing it twice undoes it, like turning a light switch on and off again. ✓
Watch out. Two reflections in the same line cancel. Two reflections in different lines do not (see section 21.6).
Method. A reflection does not change the size or shape of a figure.
- Recall what a reflection preserves. The image is congruent to the object: all lengths, angles and areas are unchanged.Congruent shapes have the same area.
- Write the area. Nothing changes.Area = 18 cm2
Check. Folding along the mirror line would put the two shapes exactly on top of each other. ✓
Watch out. Only enlargements change the size of a shape.
Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.
Method. To reflect a shape, reflect every vertex over the mirror line y = x + 1, then join the image points in the same order.
- Draw the mirror line. The line y = x + 1 is a diagonal line.Draw it through two points that fit the equation.
- Reflect each vertex of A in turn. Reflect (a, b) in y = x + 1: the new point is (b − 1, a + 1).(1, 2) → (1, 2)
(4, 2) → (1, 5)
(3, 4) → (3, 4)
(1, 4) → (3, 2) - Plot the image points and join them up in the same order to make shape B. The image has the same size and shape as the object but is the other way round, like a mirror image.B has vertices at (1, 2), (1, 5), (3, 4), (3, 2).
Check. The mirror line should be exactly half-way between a point and its image. The midpoint of (4, 2) and (1, 5) is (4 + 1) ÷ 2, (2 + 5) ÷ 2 = (2.5, 3.5), which lies on the line y = x + 1. ✓
Watch out. Reflecting in y = x swaps the coordinates; it does not change any signs. (Changing signs is for y = −x.)
Method. A reflection in a line has two features: the line joining a point to its image is at right angles to the mirror line, and the mirror line passes through the midpoint. Check both for (a, b) and (−b, −a).
- Find the midpoint. Add the coordinates and halve.Midpoint = ((a − b) ÷ 2, (b − a) ÷ 2)
- Check the midpoint is on y = −x. The y-coordinate must be the opposite of the x-coordinate.(b − a) ÷ 2 = −(a − b) ÷ 2, so the midpoint is on the line.
- Find the gradient of the line joining the points. Gradient = change in y ÷ change in x.(−a − b) ÷ (−b − a) = 1 (when a + b is not 0)
- Check the lines are perpendicular. The line y = −x has gradient −1, and 1 × (−1) = −1, so the lines are at right angles.Both features hold, so (−b, −a) is the reflection.
Check. Test with (a, b) = (2, 1): the rule gives (−1, −2). The midpoint is (0.5, −0.5), which is on y = −x. ✓
Watch out. If a + b = 0 the point is on the mirror line already and does not move.