Why no more: a line of symmetry of a square must pass through its centre and must send corners to corners. There are only 4 such lines — two through the midpoints of opposite sides, and two through opposite corners.
Q2.
Thus, figures can have multiple lines of symmetry. The figures below also have multiple lines of symmetry. Can you find them all?
Answer
Yes. All three figures on page 221 have more than one line of symmetry.
Figure
Lines of symmetry
Angles of symmetry
The red snowflake (Koch snowflake)
6
60°, 120°, 180°, 240°, 300°, 360°
The yellow flower with 6 petals
6
60°, 120°, 180°, 240°, 300°, 360°
The green knot design (like a looped square)
4
90°, 180°, 270°, 360°
A six-pointed figure has 6 lines of symmetry — 3 through opposite points and 3 through opposite notches. The snowflake and the 6-petal flower work exactly like this.
How to count quickly: if a design is built by repeating one piece n times around a centre and the piece itself is not tilted, the design has n lines of symmetry and n angles of symmetry.
Q3.
We saw that the diagonal of a square is also a line of symmetry. Let us take a rectangle that is not a square. Is its diagonal a line of symmetry? First, see the rectangle and answer this question. Then, take a rectangular piece of paper and check if the two parts overlap by folding it along its diagonal. What do you observe?
Answer
No. The diagonal of a rectangle is not a line of symmetry.
A rectangle has only 2 lines of symmetry (green). The diagonal (red) is not one of them.
When you fold a rectangular sheet along its diagonal, the two triangles do not cover each other — one long edge sticks out on one side and one short edge sticks out on the other.
Why it happens: a fold along the diagonal would have to send the long side (say 8 cm) onto the short side (say 5 cm). Folding cannot change a length, so 8 cm can never land on 5 cm. In a square all four sides are equal, which is exactly why its diagonals do work.
So a rectangle that is not a square has exactly 2 lines of symmetry: the line through the midpoints of the two long sides and the line through the midpoints of the two short sides.
Q4.
What if we reflect along the diagonal from A to C? Where do points A, B, C and D go? What if we reflect along the horizontal line of symmetry?
Answer
The square is labelled A (top left), B (top right), C (bottom right) and D (bottom left).
The square ABCD with the vertical line (red), the horizontal line (green) and the diagonal AC (purple).
Reflection in the diagonal AC:
A stays at A C stays at CB goes to DD goes to B
A and C lie on the mirror line, so they do not move. B and D are on opposite sides of it, at equal distances, so they simply change places.
Reflection in the horizontal line of symmetry:
A ↔ D B ↔ C
The top edge falls on the bottom edge, so A takes the position of D, D takes the position of A, B takes the position of C and C takes the position of B.
Mirror line
A goes to
B goes to
C goes to
D goes to
Vertical line
B
A
D
C
Horizontal line
D
C
B
A
Diagonal AC
A
D
C
B
Diagonal BD
C
B
A
D
Q5.
Ink Blot Devils: Take a piece of paper. Fold it in half. Open the paper and spill a few drops of ink (or paint) on one half. Now press the halves together and then open the paper again. • What do you see? • Is the resulting figure symmetric? • If yes, where is the line of symmetry? • Is there any other line along which it can be folded to produce two identical parts?
Answer
What you see: the ink spreads and prints a copy of itself on the other half. You now see two identical blots, one on each side of the crease — often they look like a butterfly, a bat or a devil's face. That is why it is called an ink blot devil.
Is it symmetric?Yes. Whatever the blot's shape, the two halves are mirror halves of each other.
Where is the line of symmetry? Exactly along the crease — the line you folded the paper on. The crease is the mirror.
Any other line? Usually no. A random blot has just this one line of symmetry. You get a second line only if the ink drops themselves were placed symmetrically about a line at right angles to the crease.
Why it works: pressing the halves together carries every ink point to the point directly opposite it, at the same distance from the crease, on the other side. That is exactly what a reflection does.
Try This: fold the paper twice (once vertically and once horizontally), drop ink in the quarter, press, and open. Now the pattern has 2 lines of symmetry and looks the same after a half turn.