Counting lines of symmetry

Let us count lines of symmetry.

• A triangle with equal sides and equal angles (an equilateral triangle) has 3 lines of symmetry: one from each corner to the middle of the opposite side.
• A hexagon with equal sides and equal angles (a regular hexagon) has 6: three lines join opposite corners, and three more join the middles of opposite sides.

See the pattern? A shape with all sides equal and all angles equal has as many lines of symmetry as it has sides: the square has 4, a regular pentagon 5. When you count, count each line once: a line drawn right across the figure is one line, not two.

Shapes that are not regular can have fewer. A rhombus (four equal sides, but angles that are not 90°) has 2, along its diagonals. Turning a figure does not change its lines of symmetry. A square tilted so that it stands on one corner is still a square, with 4 lines of symmetry.

The book asks you to draw triangles with exactly one line of symmetry, exactly three, and none.

• A triangle with two equal sides and a third side of a different length has exactly 1.
• A triangle with all three sides equal has 3.
• A triangle with all three sides of different lengths has none.

Can a triangle have exactly 2? No. Each line of symmetry of a triangle makes two of its sides equal; two such lines would make all three sides equal, and then the triangle has 3 lines, not 2.

The book also asks you to complete drawings on squared paper so that a blue line becomes a line of symmetry. For each point of the drawing, count how many squares it is from the line, then mark a point the same number of squares away on the other side, straight across. Join these new points in the same way, and the two sides become mirror halves.