Chapter 6 of 8

All 8 chapters
  1. 1Angles on a line and round a point, readunderstood
  2. 2Angles and parallel lines, readunderstood
  3. 3Triangles, readunderstood
  4. 4Angles in polygons, readunderstood
  5. 5Drawing with ruler and compasses, readunderstood
  6. 6Line symmetry and reflection, readunderstood
  7. 7Turning symmetry, readunderstood
  8. 8Solids, nets and views, readunderstood

M3.5.6 Chapter 6

Line symmetry and reflection

In how many ways can I fold a rectangle so that both halves match?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Find all the lines of symmetry of a shape
  • Draw the reflection of a shape in a mirror line

A line of symmetry cuts a shape into two halves that match. Fold the shape along the line. The two halves lie exactly on top of each other.

Draw it first

Test for a line of symmetry

  1. An isosceles triangle. Step 1: the triangle. Step 2: a dashed line from the top corner to the middle of the base, numbered 1. Step 3: the words fold along line 1. The two halves match.

    1 Draw the shape on paper.

  2. An isosceles triangle. Step 1: the triangle. Step 2: a dashed line from the top corner to the middle of the base, numbered 1. Step 3: the words fold along line 1. The two halves match.1

    2 Draw a line you think is a line of symmetry.

  3. An isosceles triangle. Step 1: the triangle. Step 2: a dashed line from the top corner to the middle of the base, numbered 1. Step 3: the words fold along line 1. The two halves match.1fold along line 1

    3 Fold along the line. If the two halves match exactly, it is a line of symmetry.

A shape can have no line of symmetry, one line, or many. We number each line so none is counted twice.

Six shapes with every line of symmetry dashed and numbered. The rectangle has 2 lines, the square 4, the parallelogram none, the kite 1, the pentagon 5 and the house 1.12rectangle1234squareparallelogram1kite12345pentagon1house
The parallelogram has no line of symmetry. Its halves match only when you turn one over.

Worked example

How many lines of symmetry does an equilateral triangle have?

  1. Each corner has a line from it to the middle of the opposite side.
  2. Fold along that line. The two halves match.
  3. The triangle has 3 corners, so it has 3 lines.
  4. An equilateral triangle has 3 lines of symmetry.

Answer: 3

An equilateral triangle with its three lines of symmetry dashed and numbered 1, 2 and 3. Each line goes from a corner to the middle of the opposite side.123
Three lines, one from each corner.

A line of symmetry works like a mirror. The shape on one side is the reflection of the shape on the other. The reflection is called the image.

Draw it first

Draw a reflection

  1. A grid with a dashed mirror line and a triangle ABC to its left. Step 2: dashed lines go from A, B and C straight across the mirror line. Step 3: the image A', B' and C' is drawn the same distance from the mirror line on the other side. The grid is numbered along the bottom and the left side.0123456789100123456mirror lineABC

    1 Draw the shape and the mirror line on a grid.

  2. A grid with a dashed mirror line and a triangle ABC to its left. Step 2: dashed lines go from A, B and C straight across the mirror line. Step 3: the image A', B' and C' is drawn the same distance from the mirror line on the other side. The grid is numbered along the bottom and the left side.0123456789100123456mirror lineABC

    2 From each corner, draw a line straight across the mirror line.

  3. A grid with a dashed mirror line and a triangle ABC to its left. Step 2: dashed lines go from A, B and C straight across the mirror line. Step 3: the image A', B' and C' is drawn the same distance from the mirror line on the other side. The grid is numbered along the bottom and the left side.0123456789100123456mirror lineABCA'B'C'

    3 Mark each new corner the same distance from the mirror line. Join them to make the image.

Each corner of the image is as far from the mirror line as its matching corner. The image is the same size and shape.

A point such as (2, 3) means 2 across and 3 up. The mirror line x = 5 is the upright line through 5 on the bottom scale. The line y = 3 lies flat through 3 on the side scale.

Worked example

Reflect the triangle with corners A(1, 1), B(4, 1) and C(2, 4) in the mirror line x = 5.

  1. A is 5 − 1 = 4 squares left of the mirror line.
  2. So A' is 4 squares to the right, at 5 + 4 = 9.
  3. B is 1 square left of the line, so B' is at 5 + 1 = 6.
  4. C is 3 squares left of the line, so C' is at 5 + 3 = 8.
  5. The heights do not change. The image has corners (9, 1), (6, 1) and (8, 4).

Answer: (9, 1), (6, 1) and (8, 4)

A grid numbered 0 to 10 along the bottom and 0 to 6 on the left. A dashed mirror line stands at 5. The triangle ABC is hatched on the left. Its image A', B' and C' is dotted on the right. Dashed lines join each corner to its image across the mirror line.0123456789100123456mirror lineABCA'B'C'
A is 4 squares from the mirror line and A' is 4 squares on the other side.

See it three ways

With things

Stand a small mirror upright on a line drawn on paper. Put a pencil on one side of the line. Look in the mirror. The pencil seems to stand the same distance behind the mirror.

As a picture

A grid with a dashed mirror line. A triangle with corners P, Q and R sits on the left of the line. Its reflection with corners P', Q' and R' is the same distance on the right of the line. Dashed lines join the corners across the mirror line.mirror linePQRP'Q'R'

In symbols

A point at 1 is 4 squares from the mirror at 5. Its image is 4 squares the other side, at 5 + 4 = 9.

Try it

Make a symmetrical shape

You need: a sheet of thin paper, a pencil.

  1. Fold the paper in half and open it out. The fold is the mirror line.
  2. On one side, draw half of a shape. Let it touch the fold.
  3. Fold the paper again. Hold it up to a window.
  4. Trace the lines you can see through the paper onto the other side. Open it out.

What you will see: You see the whole shape. The two halves match, and the fold is its line of symmetry.

If you cannot try it, this is what happens.
A house shape with one dashed line down the middle, numbered 1. Both halves are the same.1

A wrong idea many people have

A diagonal of a rectangle is a line of symmetry.

Why it feels right: It cuts the rectangle into two halves of the same size and shape.

The test that breaks it: Cut out a rectangle of paper. Fold it along a diagonal. The corners do not meet, because the two halves do not lie on top of each other.

What is true: A line of symmetry gives two halves that fold onto each other exactly. Halves of equal size are not enough.

Key words

line of symmetry
a line that cuts a shape into two halves that fold exactly onto each other.
mirror line
the line a shape is reflected in.
reflection
the shape you see on the other side of a mirror line.
image
the shape that a reflection makes.

Back to your guess

The answer is A: Two ways.

You can fold a rectangle across the middle in two ways: top to bottom and left to right. A fold along a diagonal does not work. The two halves do not lie on top of each other.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. How many lines of symmetry does a square have?

    A square with no lines drawn on it.
    Tap your answer.

    1. Not this one. You found the two lines through the middle of opposite sides. There are two more, along the diagonals. Go to M3.5.6. Try another.

    2. Not this one. There is more than one fold that makes the halves match. Try the middle of the sides and the diagonals. Go to M3.5.6. Try another.

    3. Yes, that is right. A square folds onto itself across the middle two ways and along each diagonal. That is 2 + 2 = 4 lines.

    See the answer

    4

    A square folds onto itself across the middle two ways and along each diagonal. That is 2 + 2 = 4 lines.

    • If you chose A: You found the two lines through the middle of opposite sides. There are two more, along the diagonals. Go to M3.5.6.
    • If you chose B: There is more than one fold that makes the halves match. Try the middle of the sides and the diagonals. Go to M3.5.6.

  2. Which of these shapes has no line of symmetry?

    Three shapes with their names: a rectangle and a kite on top, and a parallelogram below them.rectanglekiteparallelogram
    Tap your answer.

    1. Not this one. A rectangle folds onto itself across the middle, top to bottom and left to right. Go to M3.5.6. Try another.

    2. Yes, that is right. No fold makes the two halves of a parallelogram lie on top of each other. The rectangle has 2 lines and the kite has 1.

    3. Not this one. A kite has one line of symmetry, from the top corner to the bottom corner. Go to M3.5.6. Try another.

    See the answer

    Parallelogram

    No fold makes the two halves of a parallelogram lie on top of each other. The rectangle has 2 lines and the kite has 1.

    • If you chose A: A rectangle folds onto itself across the middle, top to bottom and left to right. Go to M3.5.6.
    • If you chose C: A kite has one line of symmetry, from the top corner to the bottom corner. Go to M3.5.6.

  3. The triangle ABC is reflected in the mirror line x = 5. A is at (2, 3). Where is A'?

    A grid numbered 0 to 10 along the bottom and 0 to 7 on the left. A dashed mirror line stands at 5. The hatched triangle ABC is on the left of it. The image is not drawn.01234567891001234567mirror lineABC
    Tap your answer.

    1. Yes, that is right. A is 5 − 2 = 3 squares left of the mirror line. A' is 3 squares to the right, at 5 + 3 = 8. The height stays 3.

    2. Not this one. The mirror line is upright, so only the first number changes. The second number stays 3. Go to M3.5.6. Try another.

    3. Not this one. A is 3 squares from the mirror line, and A' is 3 squares on the other side. 5 + 3 is 8, not 7. Go to M3.5.6. Try another.

    See the answer

    (8, 3)

    A is 5 − 2 = 3 squares left of the mirror line. A' is 3 squares to the right, at 5 + 3 = 8. The height stays 3.

    • If you chose B: The mirror line is upright, so only the first number changes. The second number stays 3. Go to M3.5.6.
    • If you chose C: A is 3 squares from the mirror line, and A' is 3 squares on the other side. 5 + 3 is 8, not 7. Go to M3.5.6.

  4. How many lines of symmetry does a regular hexagon have?

    A regular hexagon.
    Tap your answer.

    1. Not this one. You found the lines through opposite corners. There are also three lines through the middles of opposite sides. Go to M3.5.6. Try another.

    2. Not this one. That is two for each side. A line of symmetry goes right across the shape, so each line is counted once. Go to M3.5.6. Try another.

    3. Yes, that is right. Three lines join opposite corners and three join the middles of opposite sides. 3 + 3 = 6.

    See the answer

    6

    Three lines join opposite corners and three join the middles of opposite sides. 3 + 3 = 6.

    • If you chose A: You found the lines through opposite corners. There are also three lines through the middles of opposite sides. Go to M3.5.6.
    • If you chose B: That is two for each side. A line of symmetry goes right across the shape, so each line is counted once. Go to M3.5.6.

  5. The triangle is reflected in the mirror line y = 3. The corner P is at (1, 4). Where is P'?

    A grid numbered 0 to 7 along the bottom and 0 to 7 on the left. A dashed mirror line lies across it at 3. A hatched triangle with corners P, Q and R is above the mirror line. The image is not drawn.0123456701234567mirror linePQR
    Tap your answer.

    1. Not this one. You moved P up by 1 again. The image is on the other side of the mirror line, below it. Go to M3.5.6. Try another.

    2. Yes, that is right. P is 4 − 3 = 1 square above the mirror line. P' is 1 square below, at 3 − 1 = 2. The first number stays 1.

    3. Not this one. That is on the mirror line itself. P is 1 square above it, so P' is 1 square below: 3 − 1 = 2. Go to M3.5.6. Try another.

    See the answer

    (1, 2)

    P is 4 − 3 = 1 square above the mirror line. P' is 1 square below, at 3 − 1 = 2. The first number stays 1.

    • If you chose A: You moved P up by 1 again. The image is on the other side of the mirror line, below it. Go to M3.5.6.
    • If you chose C: That is on the mirror line itself. P is 1 square above it, so P' is 1 square below: 3 − 1 = 2. Go to M3.5.6.

  6. A regular polygon has 7 sides. How many lines of symmetry does it have?

    See the answer

    7

    A regular polygon has one line of symmetry for each side, so 7 sides give 7 lines.

  7. From an earlier chapter

    What is each angle of a regular octagon?

    See the answer

    135°

    An octagon makes 8 − 2 = 6 triangles, so the angle sum is 6 × 180° = 1 080°. Then 1 080° ÷ 8 = 135°.

Keep it

Copy this into your exercise book.

A line of symmetry gives two halves that fold onto each other. A reflection puts every point the same distance from the mirror line on the other side. A regular polygon has as many lines of symmetry as sides.

Draw: A triangle on a grid, a dashed mirror line, and its reflection. Join each corner to its image with a dashed line.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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