Lines of Symmetry

Worked example

1. 1

   $\text{regular octagon: }8\text{ sides}$ — Regular means all sides and all angles are equal, so every corner and every side has a matching partner.

2. 2

   $\text{corner to opposite corner: }8÷2=4$ — Each line of this kind joins two opposite corners, so it uses up 2 of the 8 corners.

3. 3

   $\text{middle of a side to the opposite middle: }8÷2=4$ — Each line of this kind joins the middles of two opposite sides, so it uses up 2 of the 8 sides.

4. 4

   $4+4=8\text{ lines}$ — Add the two kinds. The total equals the number of sides.

Worked example

1. 1

   $\text{corners across the bottom: }1\text{ and }5$ — The two outer corners of the trapezium.

2. 2

   $(1+5)÷2=3$ — The middle of two numbers is their mean. The mirror line must pass through here.

3. 3

   $5-3=2,3-1=2$ — Check the bottom corners: each is 2 squares from the line, one on each side.

4. 4

   $\text{top corners: }4-3=1,3-2=1$ — Check the top corners at 2 and 4: each is 1 square from the line, one on each side.

5. 5

   $\text{draw the upright line at }3$ — Every corner has a partner the same distance away on the other side, so this line works.

Worked example

1. 1

   $\text{6 columns, mirror line between columns 3 and 4}$ — So column 3 swaps with column 4, column 2 with 5 and column 1 with 6.

2. 2

   $\text{column }c\to 7-c$ — Each pair of columns adds up to 7, so the partner of column c is column 7 − c.

3. 3

   $\text{row 1, column 2}\to \text{row 1, column }7-2=5$ — A vertical mirror swaps left and right only, so the row stays the same.

4. 4

   $\text{row 2, column 3}\to \text{row 2, column }7-3=4$ — Same move for the next shaded square.

5. 5

   $\text{row 3, column 1}\to \text{row 3, column }7-1=6$ — Same move again.

6. 6

   $\text{shade the 3 new squares}$ — Check by folding: every shaded square now lands on a shaded square.

Worked example

1. 1

   $(2,2)\text{ and }(2,3)$ — Look for a pair that already mirror each other. They are side by side in the same row.

2. 2

   $\text{try the line between columns 2 and 3}$ — Test this as the line. Columns 2 and 3 swap, and 1 and 4 swap: c becomes 5 − c.

3. 3

   $(1,1)\to (1,5-1)=(1,4)$ — The square at row 1, column 1 has no partner yet. Its partner is in the same row, column 5 − 1 = 4.

4. 4

   $\text{shade }(1,4)$ — Now every shaded square has a partner. A flat line would need squares in row 4 and neither slanted line works, so there is exactly one line.