PrepShorts · Study sheet · Class 6 Mathematics · Chapter 9, Symmetry
Chapter 9 · Symmetry
Why a figure can have several lines of symmetry
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Symmetry is a count, not a yes or no. Fold a square four different ways and every one of them works.
The idea
Symmetry is not a yes-or-no property, and the reason is that each successful fold is a separate fact about the figure. So the honest answer to "is it symmetric?" is a number, and that number is fixed by the figure's measurements rather than by how neatly it is drawn: a square and a rectangle look equally regular, yet the square accepts a diagonal fold and the rectangle refuses it, because a fold along a diagonal demands that the two sides meeting at that corner be equal.
What you should be able to do
- Find every line of symmetry of a square by folding, and say how many there are
- Explain why a square's diagonal is a line of symmetry and a non-square rectangle's diagonal is not
- State the number of lines of symmetry of a regular polygon with a given number of sides, and justify it
- Draw triangles with exactly three, exactly one, and no lines of symmetry
- Argue that no triangle has exactly two lines of symmetry
- Produce figures with a curved boundary having a stated number of lines of symmetry
- Count the lines of symmetry of a traditional design or a paper cut-out
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| line of symmetry | a line the figure can be folded along so the parts land on each other | §9.1, p.219 — printed in bold and defined there |
| multiple lines of symmetry | the situation where more than one such line exists | §9.1, p.221 and Summary, p.241 — stated on both pages |
| diagonal | the line joining two opposite corners of a four-sided figure | §9.1, pp.220–221 — used repeatedly in the square and rectangle passage |
| square | a four-sided figure whose sides are all equal and whose angles are all equal | §9.1, p.220 — the worked case of the section |
| rectangle | a four-sided figure with equal angles whose adjacent sides may differ | §9.1, p.221 — the contrast case |
| regular polygon | a many-sided figure whose sides are all equal and whose angles are all equal | §9.2, p.239, Q8 — named there and carried over from Chapter 1, §1.5, Table 3, p.10 |
| hexagon | a six-sided figure | §9.1, p.227, Q6c — set as an exercise, its sides all equal and its angles all equal |
| kolam | a South Indian floor design drawn around a dot grid | §9.1, p.228 — printed in Q8 |
| equilateral | having all three sides equal | not printed in this chapter — the book describes such a triangle by its equal sides and angles instead (§9.1, p.226, Q6b) |
| isosceles | having exactly two sides equal | not printed in this chapter — the explanation's word for the one-fold-line triangle of Q9a |
| fold count | the number of lines of symmetry a figure has | an added shorthand; the book counts without naming the count |
Where people slip up
- "If one diagonal is a line of symmetry, every diagonal is." The square and the rectangle sit on the same page of the book precisely to break this.
- "A rectangle has four lines of symmetry, like a square." The two mid-lines work; neither diagonal does. Two, not four.
- "More sides always means more symmetry." True only for regular polygons. An irregular hexagon can have none. The word regular is doing all the work in section 8.
- "Turning the figure changes its lines of symmetry." The square standing on a corner (Q6a, p.226) still has four; only their directions on the page move.
- "Exactly two lines of symmetry is impossible." It is impossible for a triangle, which is what Q9 asks. A rectangle has exactly two. Keep the scope of the impossibility explicit or students over-generalise it.
- "Curved figures cannot have lines of symmetry." Q10 exists to refute this; the disc, the oval and the rosette are all fold-testable.
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Worked answers to this chapter’s exercises · this video explains Figure it Out · 2 Q6, Figure it Out · 2 Q7, Figure it Out · 2 Q8, Figure it Out · 2 Q9, Figure it Out · 2 Q10, Figure it Out · 4 Q8
Transcript1,310 words
Here is a square, and here is a line down the middle of it. Fold along that line and the two halves land on each other exactly. So the line passes the test. And it is tempting to stop there. The square is symmetric. Done. But look at what has actually been established. One line works. That is one fact about this square. Nothing in it says there is not a second line that works as well, or a third.
Each line that passes is a separate fact about the figure, and stopping at the first one throws the rest away. So the real question is not whether the square is symmetric. It is how many lines pass. Take a sheet of paper cut square and find out. The trick is to open it again after every fold, so the creases build up on one sheet. Fold it upright, down the middle, and open it. There is a crease, and the halves landed exactly.
Fold it flat, across the middle, and open it. A second crease. Now fold it corner to corner. Open it. A third. And the other way, corner to corner again. A fourth. Four folds, four creases, and every one of them landed the halves exactly on each other. So this square has at least four lines of symmetry, and all four run through its centre. At least four. Is there a fifth?
Here is the argument that settles it. When the paper folds, every corner has to land somewhere, and where it lands has to be another corner. Otherwise there is paper sticking out. So a fold line has to send the four corners to the four corners. Try a line at some other angle. This corner comes down here, and there is no corner there. Paper over the edge. Only four lines send corners to corners: the two through opposite sides, and the two through opposite corners.
Four. Not four so far. Four, and that is the end of it. Now a rectangle. Eight centimetres long, three centimetres tall. It looks every bit as tidy as the square. The same right angles, the same straight sides, drawn with the same ruler. Fold it upright, down the middle. That works. Fold it flat, across the middle. That works too. So two already, and they are the same two the square had.
The square also had two more, along its diagonals. So here is the diagonal of this rectangle. Before I fold along it, decide what you think is going to happen. It is worth committing to an answer, because most people commit to the wrong one. Fold. A strip of paper hangs out past the edge, and there is a lot of it. Look at the corner the fold started from. Two sides meet there. One of them is eight centimetres long and the other is three.
The fold has to lay one of those two sides down onto the other. Eight cannot land on three. So the long side swings over, crosses the top edge about three and a half centimetres along, and carries straight on. Three and a half centimetres of it finish outside the paper altogether. The diagonal does not nearly work. It fails, and it fails by a lot. Which settles the rectangle. Its two middle lines work. Neither diagonal does.
Two lines of symmetry. Not four. And now look back at the square, and ask why its diagonals worked. Because at the corner where that fold starts, the two sides meeting there are the same length. Eight and eight, or three and three. That is the whole difference between these two figures. Not the neatness. Not the right angles. One measurement. A square is a rectangle whose sides happen to be equal, and that single equality is worth two extra fold lines.
Three more, counted rather than admired. A snowflake with six spikes. Fold through a spike and it works. Fold between two spikes and that works as well. Keep going round. Six lines. A square knot with a loop at each corner. Upright, flat, and both diagonals. Four. A star with eight points, made of two squares laid across each other. Through each point, and between each pair of points. Eight.
Six, four, eight. Not one of those numbers came from how nice the figure looks. Every one of them was folded for, line by line, and stopped when the lines ran out. There is a pattern coming, and it is worth stating carefully. A triangle with all three sides equal and all three angles equal has three lines of symmetry. A square has four. A five-sided figure of the same kind has five. Six sides, six.
Seven, eight, nine, ten. Every one of them has exactly as many fold lines as it has sides. But the words all equal are carrying that sentence. Here is a six-sided figure whose sides are all different lengths. It has none. Not a single one. More sides does not buy you more symmetry. Equal sides and equal angles do. Triangles deserve a moment of their own, because they can only manage three answers.
All three sides equal. Three fold lines, one through each corner. Exactly two sides equal. One fold line, through the corner where those two meet. All three sides different. None at all. Three, one, or zero. Draw as many as you like. Long thin ones, flat ones, ones with a corner pushed right out to the side. You will get three, or one, or none, every single time. And you might notice what is missing from that list. Two.
Here is why two is impossible, and the argument is short. Take any fold line of a triangle. A triangle has three corners, and three is an odd number, so the fold cannot pair them all off. One corner has to sit on the line itself, and the other two swap places. Which means the two sides meeting at that fixed corner swap as well. So those two sides are equal.
Every fold line of a triangle makes one pair of its sides equal. Now suppose there were a second one. It holds a different corner still, so it makes a different pair equal. Two pairs, out of three sides, and the two pairs have to share a side. All three come out equal. And a triangle with all three sides equal has three fold lines, not two. So two is not a near miss. It cannot happen at all.
One more thing to clear up, because it trips people. Nothing in the test ever asked for straight edges. Here is half a disc. One round edge and one straight one. Fold it upright and it works. Fold it any other way and it does not. One line. Here is an oval. Upright works. Flat works. Nothing else does. Two. And a flower of four identical petals. Through opposite petals, twice. Between the petals, twice more. Four.
Curved edges fold exactly as well as straight ones. The test never cared what the boundary was made of. So what is this number actually telling you? Not how carefully the figure was drawn. The square and the rectangle came off the same ruler. Not which way up it is sitting. Turn a figure and its fold lines turn with it. The count does not move. What the number measures is how much of the figure repeats. How many different ways it can be laid down on top of itself.
Stretch a square by a single millimetre and two of its four fold lines are gone on the spot. Nothing about it looks different. The count is different. That is why the honest answer is a number and not a yes. Yes and no cannot tell those two squares apart. Four and two can.
Where this fits
Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.
Builds on
- A line of symmetry is a fold that makes the halves coincideClass 6 · Ch 9, Symmetry
- Shapes come in sequences too, with rules of their ownClass 6 · Ch 1, Patterns in Mathematics
Comes up again in
- Symmetry as reflection: the fold line acts as a mirrorClass 6 · Ch 9, Symmetry
- Fold, blot, cut, punch: making a figure that is symmetric by constructionClass 6 · Ch 9, Symmetry
- Line symmetry and rotational symmetry are independent of each otherClass 6 · Ch 9, Symmetry