Constructing the symmetric image of a figure

Point-by-point construction method

To construct the symmetric point A' of a point A with respect to a line (d):

1. Draw the perpendicular to (d) passing through A.
2. Measure the distance between A and the point of intersection with (d); let’s call this point H.
3. Transfer the same distance from the other side of (d) onto the perpendicular line: this gives A’ such that AH = HA’.

To construct the reflection of an entire figure (triangle, line segment, polygon), repeat this method for each vertex, then join the images in the same order.

Using graph paper

On graph paper, it’s simpler: count the number of squares between each point and the axis, then mark out the same number of squares on the other side.

Example: if point A is 3 squares to the left of the vertical axis (d), then A’ will be 3 squares to the right of (d), at the same height.

Using tracing paper

You can also fold a sheet of tracing paper along the axis: the shape obtained after folding the tracing paper along the axis directly gives the symmetrical image.

Common pitfall

Do not confuse ‘same distance from the axis’ with ‘same distance from any point’: the figure must be perpendicular to the axis; otherwise, the resulting figure is not the true symmetrical image.

Furthermore, the order of the vertices must be respected: if triangle ABC is traversed in a clockwise direction, its symmetrical figure A’B’C’ will be traversed in the opposite direction.