Line symmetry
When can you fold a picture so that one half lands exactly on the other half?
Where you are 11 in Transformations
Why this exists
In Fiadh’s art class everyone is making butterfly pictures. Fiadh paints one wing in wet paint, folds the paper along a pencil line, presses, and opens it. A second wing has appeared, an exact copy, flipped over. Her picture now has a special property: fold it on that crease and every part of one wing lands on the matching part of the other.
Many things share that property: letters, leaves, flags, a square. It lets you describe a shape with half the work, because half of it and the fold line tell you the rest. It also lets you check a drawing: if a design is meant to match itself, every point on one side must have a partner on the other, and partners can be checked by counting squares.
In general, a symmetry of a figure is a move that leaves it looking exactly as it did. This lesson is about the folding kind. The turning kind is rotational symmetry.
Definition
A line of symmetry of a figure is a line you can fold the figure along so that the two sides match exactly. Every point of the figure on one side lands on a point of the figure on the other side, and nothing is left over on either side. A figure with at least one line of symmetry has line symmetry. A line of symmetry is sometimes called a mirror line: stand a mirror on it and the mirror shows the missing half.
When the figure is folded on a line of symmetry, each point P lands on a point P' (read “P prime”). The two are matching points. A point on the line stays where it is, so it is its own match.
Intuition
Folding does not stretch or shrink the paper. So in Figure 1, P and P' are the same distance from the fold line, on opposite sides of it. The dotted segment that joins them crosses the fold line at a right angle, so it is perpendicular to the fold line.
That gives two ways to test a line.
- The fold test. Trace the figure, cut it out, and fold on the line. The two sides must match exactly, edge on edge and corner on corner.
- The matching-point test. On a grid with an up-and-down fold line, take each corner on one side and count the squares straight across to the line. Its partner must be the same number of squares on the other side, in the same row. With an across fold line, count up and down, and the partner is in the same column.
“Exactly” is the whole point. A drawing that is nearly the same on both sides does not have line symmetry. Neither does one that splits into two halves of the same size and shape but needs a turn, not a fold, to lay one on the other (Example 2).
Results
Matching points. If a line is a line of symmetry, every point of the figure off the line has a matching point on the other side, the same distance from the line. The segment joining them is perpendicular to the line and cut in half by it. A point on the line is its own match.
How many lines. A figure can have 0, 1, 2 or more lines of symmetry, or infinitely many. Some figures that come up often:
| Figure | Lines of symmetry |
|---|---|
| square | 4 |
| rectangle that is not a square | 2 |
| parallelogram that is neither a rectangle nor a rhombus | 0 |
| regular polygon with n sides | n |
| circle | infinitely many: every line through the center |
A regular polygon has all its sides the same length and all its angles the same size, like a square or a stop sign. Polygon names say how many sides it has.
Why it is true
Matching points. Fold on the line, and P lands on P'. The point Q where segment PP' crosses the fold line does not move. The paper does not stretch, so QP and QP' have the same length: the line cuts the segment in half. The fold also lays the angle on one side of Q exactly onto the angle on the other side, so those two angles are equal. Together they make a straight angle of 180^\circ, so each is 90^\circ.
Regular polygons. A fold that matches a polygon must put corners onto corners and sides onto sides. So a line of symmetry either runs through a corner, or crosses a side at its middle and swaps that side’s two ends.
In a regular polygon every such line works. Fold through a corner and the center: the two sides at that corner are the same length and meet at the same angle as every other pair, so they land on each other, then the next two sides do, and so on all the way round.
Now count. With an odd number of sides, such as 5, a line through a corner and the center comes out at the middle of the opposite side. There is one line for each corner, so 5 lines. With an even number, such as 8, corners come in opposite pairs, giving 4 lines from corner to corner, and sides come in opposite pairs, giving 4 more lines from middle to middle: 8 in all. Either way a regular polygon with n sides has n lines of symmetry.
The circle. Every point of a circle is the same distance from the center. Fold on any line through the center and each point of the circle lands on a point that is also that distance from the center, so it lands on the circle. There is no end to the lines you can draw through one point.
Worked examples
Examples 1 to 3 are the first day’s work: testing a fold line and completing a figure. Examples 4 to 7 are the second day’s: finding every line of symmetry a figure has.
Checking a butterfly by counting squares
Fiadh draws the butterfly in Figure 2. She says the dashed line is a line of symmetry. Is she right?
The fold line goes up and down, so matching points sit in the same row, the same number of squares from the line on each side. Pair up the corners row by row.
| Row of | Left of the line | Right of the line | Match? |
|---|---|---|---|
| A and E | 4 squares | 4 squares | yes |
| B and F | 6 squares | 5 squares | no |
| C and G | 2 squares | 2 squares | yes |
| D and H | 4 squares | 4 squares | yes |
The two points where the outline touches the line are their own matches. But folding would put B one square past F, on bare paper. One pair that fails is enough: the dashed line is not a line of symmetry. Moving F one square to the right, or B one square to the right, would make Fiadh’s claim true.
Two equal halves that do not fold together
Senan cuts out rectangle ABCD, 6 units wide and 3 units tall. The diagonal AC splits it into two triangles of exactly the same size and shape. Is AC a line of symmetry?
Do the fold test. Folding on AC carries triangle ABC over to the other side, as the dashed triangle in Figure 3 shows. Corner B lands at B', above the top edge, outside the rectangle. The two sides do not match, so AC is not a line of symmetry.
Why it fails: for the two sides to match, the fold would have to put B on D, the only corner on the other side. Folding does not stretch the paper, so B' is as far from A as B is, 6 units. But D is only 3 units from A, so B cannot land on D.
The two triangles really are the same size and shape. You could cut one out, turn it halfway round, and lay it on the other. A line of symmetry needs more than that: the halves must match when folded, with no turning.
Completing half a butterfly
Siobhan draws the left half of a butterfly (Figure 4). Finish the drawing so that the dashed line is a line of symmetry.
Each corner needs a partner in its own row, the same number of squares to the right of the line as the corner is to the left.
| Corner | Squares left of the line | Partner | Squares right of the line |
|---|---|---|---|
| A | 3 | A' | 3 |
| B | 5 | B' | 5 |
| C | 3 | C' | 3 |
| D | 4 | D' | 4 |
| E | 2 | E' | 2 |
The two ends of the half are on the line, so they stay put. Mark the five new corners, then join them in the same order as the old ones, from the top end round to the bottom end.
Check. Fold on the line: A meets A', B meets B', and so on, and each side drawn on the left lands on the side drawn between the partners on the right.
Lines of symmetry of four letters
Find all the lines of symmetry of the block letters A, E, H and N, drawn with straight strokes as in Figure 6.
SolutionTest each possible fold: up and down through the middle, across through the middle, and slanted.
A. An up-and-down fold through the top point swaps the two legs, and the crossbar folds onto itself. A fold across would send the point at the top down to where the feet are, so it fails. 1 line.
E. A fold across the middle bar sends the top bar onto the bottom bar, the middle bar onto itself, and the upright onto itself. An up-and-down fold would send the upright to the right side, where there is nothing. 1 line.
H. An up-and-down fold swaps the two posts. A fold across swaps the top halves of the posts with the bottom halves, and the bar folds onto itself. 2 lines.
N. An up-and-down fold swaps the two posts, but the slanted stroke runs from top left to bottom right, and after the fold it would run from top right to bottom left. A fold across fails in the same way. 0 lines.
A slanted fold fails for all four, because it would lay an upright stroke along a slanted line.
The counts belong to these drawings. In a different typeface, or in handwriting, a letter can gain or lose a line, so the letters must be drawn before they are counted.
A square, a rectangle and a parallelogram
Siobhan cuts out a square, a rectangle that is not a square, and a parallelogram that is neither a rectangle nor a rhombus. Find all the lines of symmetry of each.
SolutionSquare. Folding top onto bottom works, and so does left onto right. A fold from corner to corner also works: the two sides at that corner are the same length and meet at a right angle, so one lands exactly on the other, and the other two sides do the same. There are two diagonals. 4 lines.
Rectangle. Top onto bottom and left onto right both work. The diagonals fail for the reason in Example 2: a fold on a diagonal would have to put a corner that is a long side away onto a corner that is only a short side away. 2 lines.
Parallelogram. A fold that swaps the two ends of the top side must cross it at a right angle, as the fold line does in Figure 1. The line through the middles of the top and bottom sides runs on a slant, so it does not, and the top side does not land on itself. The line through the middles of the slanted sides is level, so it does not cross them at a right angle, and it fails the same way. A fold along a diagonal would have to lay each side onto the side next to it, and in this parallelogram neighboring sides have different lengths. These are all the folds that put corners onto corners, so there are 0 lines.
Regular polygons and the circle
How many lines of symmetry do a regular pentagon, a regular octagon and a circle have?
SolutionPentagon. It has 5 sides, an odd number. Each line through a corner and the center comes out at the middle of the opposite side. One line per corner gives 5 lines.
Octagon. It has 8 sides, an even number. Its corners make 4 opposite pairs, so 4 lines go from corner to corner. Its sides make 4 opposite pairs, so 4 more lines go from the middle of a side to the middle of the opposite side. That is 4 + 4 = 8 lines.
Circle. Every line through the center is a line of symmetry, and there is no end to those lines. A circle has infinitely many.
Check. In both polygons the count equals the number of sides, as the Results say it must.
A design with two fold lines
Siobhan draws a quarter of a design (Figure 9). Complete it so that both dashed lines are lines of symmetry. How many corners does the finished design have? Does it have any other line of symmetry?
Work one fold at a time. First reflect across the up-and-down line: A, B and C are 2, 4 and 3 squares to the right of it, so their partners go 2, 4 and 3 squares to the left, in the same rows. Now the top half is done.
Next reflect the whole top half across the line going across. Each corner gets a partner in its own column, as many squares below the line as it is above. Figure 10 shows the result.
Count corners. A, B and C are off both lines, and each now appears once in every quarter: 3 \times 4 = 12 corners. The path also touches the lines at 4 points (top, bottom, left and right), and the outline bends at each of them, so each is a corner. That makes 12 + 4 = 16.
Is a diagonal a line of symmetry? The top tip is 5 squares up and the right tip is 5 squares right, which looks hopeful. But a fold on the diagonal swaps “squares right” with “squares up”, so B, at 4 right and 5 up, would land at 5 right and 4 up. No corner is there. The design has exactly 2 lines of symmetry.
Boundaries
The match must be exact. Close is not enough. A hand drawing can only be nearly symmetric; the figure it stands for either matches exactly or does not.
Same size and shape is not enough. A line can cut a figure into two halves that match after a turn and still not be a line of symmetry (Example 2).
Everything in the figure counts. A butterfly whose outline matches but which has a spot on one wing only has no line of symmetry. The fold has to match color and markings too.
Letters depend on how they are drawn. Count the letter in front of you, not the letter in general.
Folding is not turning. N, S and Z look the same after a half turn but have no line of symmetry. A figure can have either kind of symmetry without the other.
A fold line belongs to a flat figure. A solid, such as a box, is split by a flat mirror instead of a line. That is a different idea.
Where this goes next
A fold on a line of symmetry is a reflection, a flip across a line. On a coordinate grid the matching-point rule becomes a short rule about coordinates: reflecting over the axes changes the sign of one of them. The same fold tests graphs: a graph that matches itself when folded on the vertical axis is symmetric about that axis, and later an equation can say so without drawing.
Lines of symmetry also sort shapes. The fold line of a kite is what proves its angle facts in the properties of special quadrilaterals. In a regular polygon the lines of symmetry all meet at the center, which leads to the angle at the center between neighboring corners. And designs that repeat a shape across a whole floor, matching across many lines at once, are Tessellations.
Exercises
Part 1: testing a fold line
Use Figure 11 for Exercises 1 to 4. Is the dashed line a line of symmetry? Explain by matching corners.
- Shape (a), the house.
- Shape (b), the L shape.
- Shape (c), the arrow.
- Shape (d), the square.
Part 1: matching points
- A figure has an up-and-down line of symmetry. One corner is 7 squares to the left of the line. Where is its matching corner?
- Two matching points are 10 squares apart, with an up-and-down line of symmetry between them. How far is each point from the line?
- A corner of a figure lies on its line of symmetry. Where is its matching point? Explain.
- Senan folds his butterfly picture on a crease. One spot on the left wing lands 1 square away from the matching spot on the right wing. Is the crease a line of symmetry? Explain.
Part 1: completing a figure
- Complete the shape in Figure 12. Say where A', B' and C' go. How many corners does the finished shape have?
- Complete the shape in Figure 13. Say where A', B' and C' go. How many corners does the finished shape have?
Part 2: letters
Use Figure 14. How many lines of symmetry does each letter have, as drawn?
- T
- M
- Z
- I
Part 2: shapes
How many lines of symmetry does each shape have?
- A triangle with two sides of the same length and a third side of a different length.
- A rhombus that is not a square.
- A regular decagon (ten sides).
- A triangle whose three sides all have different lengths.
- A semicircle, half of a circle cut along a line through its center.
Part 2: designs
- Fiadh folds a square sheet of paper in half, then in half again the other way, so the two creases cross at the center. She cuts notches through all the layers and unfolds it. Which lines must be lines of symmetry of her paper snowflake, and so what is the least number of lines it can have?
- Complete the design in Figure 15 so that both dashed lines are lines of symmetry. How many corners does it have, and how many lines of symmetry?
Reasoning
- Siobhan says, “A line that cuts a shape into two halves of the same size and shape is always a line of symmetry.” Give a shape and a line that show she is wrong, other than the rectangle’s diagonal.
- Can a triangle have exactly 2 lines of symmetry? Explain.
- Senan says a circle has 4 lines of symmetry, like a square. Explain why a circle has more than any number he could name.
- Fiadh says a regular polygon with 12 sides has 6 lines of symmetry, “one through each pair of opposite corners.” What has she missed, and what is the right number?
Answers
- Yes. The two bottom corners and the two corners where the roof starts are each 2 squares from the line, one on each side, in the same row. The peak is on the line.
- No. The two corners of the foot match, 2 squares each side. But the top of the L is only on the left: its top corner, 2 squares left of the line, has nothing 2 squares right of it in that row.
- Yes. For an across line, partners are in the same column. The corners 1 square above and 1 square below the line match, so do the corners 2 above and 2 below, and the tip is on the line.
- No. The line is 1 square right of the middle. The left side is 3 squares from it and the right side only 1 square, so a fold would leave 2 squares hanging over.
- 7 squares to the right of the line, in the same row. Matching points are the same distance from the line on opposite sides.
- 5 squares. The line cuts the segment between them in half, and 10 \div 2 = 5.
- It is its own matching point. Folding on a line does not move the points on the line.
- No. On a line of symmetry every point lands exactly on its matching point. A spot 1 square off means the two sides do not match.
- A' is 3 squares right of the line, in A’s row; B' is 1 square right, in B’s row; C' is 1 square right, on the bottom row. The shape has 7 corners: the top point, A, B, C, C', B' and A'. The bottom runs straight from C to C', so the point where it meets the line is not a corner.
- A' is 2 squares below the line, in A’s column; B' is 3 squares below, in B’s column; C' is 1 square below, in C’s column. The shape has 8 corners: A, B, C, their three partners, and the two ends on the line, where the outline bends.
- 1: up and down through the middle of the stem. The bar’s two halves swap.
- 1: up and down through the middle, where the two slanted strokes meet.
- 0. Like N, any fold makes the slanted stroke run the wrong way.
- 2: up and down through the stem, and across through the middle, which swaps the top bar with the bottom bar.
- 1: the line through the corner where the two equal sides meet and the middle of the third side. It swaps the two equal sides.
- 2: its two diagonals. All four sides are the same length, so a fold on a diagonal lays each side on its neighbor. The lines through the middles of opposite sides fail, because the corners are not right angles.
- 10. A regular polygon with n sides has n lines: here 5 go through opposite corners and 5 through the middles of opposite sides.
- 0. A line of symmetry of a triangle must swap two sides, so those two would have to be the same length.
- 1: the line through the center that is perpendicular to the straight edge. It swaps the two halves of the curve.
- Both creases. Every cut went through the layers on both sides of each crease, so unfolding copies each cut across both creases. The snowflake has at least 2 lines of symmetry.
- 10 corners and 2 lines of symmetry. A and B each appear in all four quarters, which gives 8 corners. The outline bends at the left and right tips, 2 more. At the top and the bottom it runs straight across the line, because the side from A to the line is level, so those points are not corners. A diagonal is not a line of symmetry: the top point is 4 squares up but the right tip is 5 squares right.
- A parallelogram that is not a rectangle, with the line through the middles of its top and bottom sides. It cuts the parallelogram into two smaller parallelograms of the same size and shape, but that line crosses the top side on a slant, not at a right angle. So a fold on it cannot swap the two ends of the top side, and the halves do not match.
- No. Each line of symmetry of a triangle goes through a corner and makes the two sides there the same length. Two lines would make two pairs of equal sides, so all three sides would be equal. Then the third corner gives a third line, so the triangle has 3 lines, not 2.
- Every line through the center is a line of symmetry, because every point of the circle is the same distance from the center. For any number of lines he names, another line through the center can be drawn between two of them, so there is always one more.
- The lines through the middles of opposite sides. 12 corners make 6 opposite pairs, giving 6 lines, and 12 sides make 6 opposite pairs, giving 6 more. The right number is 6 + 6 = 12.