Symmetry and reflections

A fold must match the whole figure

Imagine folding a paper figure so that one half lands on the other. The fold is a line of symmetry when every part of the figure matches exactly. A symmetric figure has at least one such line. Its position on the page does not decide whether it is symmetric.

Look for matching corners and edges, not just two equal areas. If one corner misses its partner, that proposed fold fails. You may still find a different fold that works.

Check your understanding 1 Look for a complete match

Which of A, B and C are symmetric? For each one you choose, sketch a possible fold. For any you leave out, explain what would fail to match. You can draw on paper.

Your figure letters and reasons

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A and C are symmetric. A folds along the line from its top side’s midpoint to the opposite corner. C folds along a horizontal line through its right-hand corner. Turning a symmetric figure does not remove its symmetry. B has no fold that matches all four corners and edges; matching one pair alone is insufficient.

A symmetric figure can have a proposed fold that fails

The rectangle below has two proposed lines. Decide which would make the whole outline match. Check where a corner would land and which sides would meet.

Check your understanding 2 Equal parts are not always mirror halves

Line n joins opposite corners and makes two equal-area triangles. Does that make n a line of symmetry? Does a failed fold mean the rectangle has no symmetry?

Your decisions and reasons

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Line m is a symmetry line: left and right corners and edges match. Line n is not. Across this diagonal, a long side would have to match a short side. Equal triangle areas do not make matching mirror halves. The failed fold n does not remove the rectangle’s working fold m.

A non-square rectangle has two symmetry lines, each through the midpoints of a pair of opposite sides. A square has four: these two and both diagonals. Its equal sides make the diagonal folds work too.

Complete a missing half by reflecting the corners

For a vertical mirror, a corner and its reflection have the same height. Count the small squares perpendicular to the mirror, then mark the same distance on the other side. A corner on the mirror stays in place. Join the reflected corners in the same order.

In this drawing, coordinates increase rightwards and down the page. Each small square spans 10 drawing units. The mirror is at x = 100. A corner at (60, 40) is four small squares to its left, so its reflection is at (140, 40). The heights agree and both are four squares from the mirror. Sliding every corner by the same amount would fail: a corner farther from the mirror needs a larger move.

Check your understanding 3 Use matching distances

A dot is three small squares to the left of a vertical mirror. How many small squares to the right is its reflection?

Small squares to the right

Check answer

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Three small squares to the right, at the same height. The dots are six squares apart, but each is three from the mirror.

Now complete the given half below. Keep the given boundary. Draw only the missing open path, starting and ending where the given path meets the mirror. The two interior corners need reflected partners.

Choose Solid line, place the missing corners in order, then Finish line. For keyboard entry, open Type coordinates instead, choose Joined points and keep Open path. Put one x, y pair on each line. Do not join across the mirror to make a second closed shape.

Try the model. You can change your answer or look at the worked model. This activity has no score.

The check compares the missing open boundary across the mirror, including its endpoints and every corner. It accepts either drawing direction. Use the small grid squares, each spanning 10 drawing units. You can also draw on paper and compare with the public answer.

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Coordinates increase to the right and down the page. Lengths use equal coordinate scales.

Points and lines in the drawing

Object	Values	Given or drawn
m	(100, 0) → (100, 200)	Given
Given boundary segment 1	(100, 20) → (60, 40)	Given
Given boundary segment 2	(60, 40) → (40, 80)	Given
Given boundary segment 3	(40, 80) → (100, 100)	Given
missing-boundary	(100, 20) → (140, 40) → (160, 80) → (100, 100); start unspecified, end unspecified	Drawn

Try a sloping mirror

The mirror direction changes how you count. A horizontal gap is perpendicular to a vertical mirror, but it is not perpendicular to a sloping mirror. For a sloping mirror, check that the line joining a corner to its reflection crosses the mirror at a right angle and is cut in half there. Squared paper can help you follow matching diagonal steps.

Complete this outline across t. Choose your own missing corners before opening the solution. The mirror runs from the top-left to the bottom-right of the grid. What happens to the horizontal and vertical steps when you reflect them?

Try the model. You can change your answer or look at the worked model. This activity has no score.

The check compares the missing open boundary across the mirror, including its endpoints and every corner. It accepts either drawing direction. Use the small grid squares, each spanning 10 drawing units. You can also draw on paper and compare with the public answer.

Show solution

Coordinates increase to the right and down the page. Lengths use equal coordinate scales.

Points and lines in the drawing

Object	Values	Given or drawn
t	(0, 0) → (200, 200)	Given
Given boundary segment 1	(40, 40) → (80, 20)	Given
Given boundary segment 2	(80, 20) → (120, 40)	Given
Given boundary segment 3	(120, 40) → (120, 120)	Given
missing-boundary	(40, 40) → (20, 80) → (40, 120) → (120, 120); start unspecified, end unspecified	Drawn

In the worked model, the corner at (80, 20) reflects to (20, 80). Horizontal steps become vertical steps and vertical steps become horizontal steps. Compare the entire completed outline, including how its edges meet the mirror.

Turn the mirror on paper

On squared paper, draw a horizontal mirror. On one side, draw an open path with two different corners between its endpoints on the mirror. Complete its missing half. Explain what stays the same and what changes compared with a vertical mirror.