Symmetry · Lesson 1 of 7
Recognising Mirror Symmetry
“Use folding and reflection to decide whether a whole figure has matching halves.”
• Recognise symmetry as a definite pattern that preserves a whole figure. • Test a proposed line of symmetry by imagining or making a fold. • Find all four reflection axes of a square and the two of a nonsquare rectangle. • Describe where labelled points move during a reflection. • Distinguish a repeated detail from symmetry of the complete design.
Look for a pattern that really matches
A butterfly often appears balanced around its body. The shapes on one wing have partners on the other wing. A flower may have similar petals arranged around its centre, while a rangoli repeats a small design in several directions. These are clues to symmetry, but a clue needs checking: the complete shape and its important details must match after the movement we propose.
Compare these patterns with an irregular cloud. You may find two bumps that look similar, yet the rest of the cloud does not follow the same arrangement. A repeated detail alone does not make the whole outline symmetric. In geometry, we look for an exact match; photographs of living things and buildings usually show an approximate version of that ideal.
A property of a figure that allows a particular reflection or rotation to leave the entire figure looking unchanged. Repeated parts fit a definite arrangement, rather than merely looking similar in isolation.
For a butterfly, a fold down the body suggests a match. For a pinwheel, turning it may suggest a match even when no fold works. We will study turns later. Start with the easier experiment: place one part of a flat drawing over another by folding the paper.
Sketch a front view of a doorway, a leaf or a building. A central tower of a gopuram and the central entrance of the Taj Mahal suggest a vertical mirror arrangement. Check the left and right details as well as the outside boundary. Treat a photograph taken from one side cautiously: the view itself can distort a symmetric structure.
The fold test
Draw a triangle with two equal sloping sides, then draw a line from its top vertex to the midpoint of the base. Folding along this line places one sloping side over the other and the two base endpoints together. A line passes the test only if all of the figure overlaps, not just one pair of points.
A line along which a figure can be folded so that its two parts overlap exactly. The parts on the opposite sides are mirror halves.
Problem
Does the vertical dotted line in each displayed figure give mirror halves?
- 1.For the triangle, compare the left and right base endpoints. They meet after the fold, and the two sloping edges lie together.
- 2.Every point along one half of the triangular boundary has a matching point on the other half. The line is therefore a line of symmetry.
- 3.For the stepped figure, the heights and steps do not match on the opposite sides. At least part of the boundary fails to overlap, so the same-looking central line is not an axis.
A puzzle-piece picture illustrates the same caution. A protruding tab on one side may face an indentation, or the upper and lower tabs may be in different places. Dividing the picture into regions that seem equally large does not guarantee mirror halves. Shape and position must agree as well.
A line through the centre is only a candidate. Equal areas on its two sides are not enough: one side must be the reflected copy of the other. Check corners, curves, holes and colours whenever those details are part of the figure.
Some figures have several mirror lines
A successful fold does not prove that you have found every axis. A square is a useful shape for a systematic search. After checking a vertical fold, open the paper again and test a horizontal fold, then each diagonal. Keep the original square available for every test so that an earlier fold does not hide a possible direction.
Problem
How many different folds divide a square into matching mirror halves?
- 1.Fold through the midpoints of the top and bottom sides. The left and right halves coincide, giving a vertical axis.
- 2.Fold through the midpoints of the left and right sides. The top and bottom halves coincide, giving a horizontal axis.
- 3.Fold along either diagonal. The two triangular halves coincide in each case. These are two more axes, making four altogether.
- 4.There are no further axes. A mirror must send each corner to a corner, and these four directions exhaust the ways that can happen while preserving the square.
Problem
A rectangle is 8 units long and 4 units high. Is a diagonal a mirror line?
- 1.A diagonal connects opposite corners and divides the rectangle into two triangles with equal areas. This fact by itself does not answer a reflection question.
- 2.If you fold along the diagonal, the long side next to a corner would have to lie over a short side. A segment of length 8 cannot coincide with a segment of length 4.
- 3.The diagonal fails. The lines through the midpoints of opposite sides do work, so a nonsquare rectangle has two axes. A square is the special rectangle whose equal side lengths allow diagonal axes too.
Stars, flowers and decorative loops can also have several axes. Try different directions through the centre, but check each petal or loop after the imagined fold. A drawing with one altered petal can lose some or all of the mirror lines of the otherwise regular pattern.
Reflection tells us where the points go
Folding gives a physical test, while reflection describes the matching positions without actually bending the paper. Think of the axis as a mirror. A point on one side has a partner the same distance from the line on the other side, measured straight across the line. Points lying on the mirror line stay where they are.
The property of a figure that matches itself when reflected across a line. Having a line of symmetry and having reflection symmetry describe the same property.
Problem
Where do A, B, C and D go when the displayed square is reflected across its vertical axis?
- 1.A is the upper-left corner and B is the upper-right corner. They are equally far from the vertical axis on opposite sides, so A goes to B and B goes to A.
- 2.The lower corners behave in the same way: D goes to C and C goes to D.
- 3.The outline still matches the original square. Labels help us follow the movement; they are temporary tracking labels, not decorations that must remain in their original places.
Across the horizontal axis, A exchanges with D and B exchanges with C. Across diagonal AC, the corners A and C lie on the axis and stay fixed, while B and D exchange. Notice the difference between the complete figure staying unchanged and every individual point staying unchanged. Most points move even when the outline matches.
Quiz
What must happen for a line to be a line of symmetry?
How many lines of symmetry does an undecorated square have?
Why is a nonsquare rectangle’s diagonal not a mirror line?
Under reflection across diagonal AC of the displayed square, which corners stay fixed?
Which observation is insufficient to prove symmetry of a cloud outline?
A point lies exactly on a reflection axis. What happens to it?
Exact overlap is the test. Equal areas or a central location alone are insufficient.
Practice Problems
- Draw a square and show all its mirror lines. Explain why a line joining adjacent side midpoints is not another axis.
- Draw a rectangle with visibly unequal length and height. Test its two diagonals as well as its two midlines.
- Sketch a butterfly-like figure with matching wings. Change one detail so that its mirror symmetry disappears.
- In the labelled square, describe the images of all four corners under horizontal reflection and under reflection across diagonal AC.
- Give a picture that has repeated details but no mirror line. Explain which parts prevent a complete match.
- Choose a building or doorway and distinguish the symmetry of its ideal front outline from the appearance of a photograph taken from one side.
- Can a line through the centre of a square fail the fold test? Draw one and justify your answer.
Horizontal reflection exchanges A with D and B with C. Reflection in AC fixes A and C and exchanges B with D.
Key Takeaways
• A mirror line preserves the complete figure when the two sides are folded together. • Symmetry includes all relevant details, not only a repeated part or equal areas. • A square has four reflection axes; a nonsquare rectangle has two. • Reflections exchange matching points across an axis and fix points on the axis. • Real-world pictures suggest symmetry; geometric tests establish an exact match.
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Next · Lesson 2
Making Symmetry by Folding and Cutting