PROPERTIES OF REFLECTIONS

A reflection is a transformation that flips a figure across a line. The line is called the line of reflection. The image and pre-image of a reflected object have some interesting mathematical properties.

The following are the three basic properties of reflections :

1. The image and the pre-image are of the same size and shape. So they are congruent.

2. The line segment which connects the corresponding parts of the image and pre-image is perpendicular to the line of reflection.

3. The distance from the line of reflection of the corresponding parts of the image and pre-image is the same.

Example :

The triangle PQR has the vertices P(1, 11), Q(1, 7) and R(3, 7). Reflect the triangle PQR across x-axis and y-axis. Determine whether the triangle PQR and its reflected objects satisfy the above mentioned properties.

Solution :

Step 1 :

Trace triangle PQR and the x- and y-axes onto a piece of paper.

Step 2 :

Fold the paper along the x-axis and trace the image of the triangle on the opposite side of the x-axis. Now, unfold the paper and label the vertices of the image P′, Q′ and R′.

Since the paper is folded along the x-axis, the line of reflection is x-axis.

Step 3 :

For the original figure, find the perpendicular distance from each point to the line of reflection.

From Point P to the line of reflection = 5 units

From Point Q to the line of reflection = 2 units

From Point R to the line of reflection = 2 units

Step 4 :

For the reflected figure, find the perpendicular distance from each point to the line of reflection.

From Point P' to the line of reflection = 5 units

From Point Q' to the line of reflection = 2 units

From Point R' to the line of reflection = 2 units

Step 5 :

From steps (3) and (4), it is clear that each point of the original figure PQR and its corresponding point on the image P'Q'R' must be at the same distance from the line of reflection.

Step 6 :

Fold the paper along the y-axis and trace the image of the triangle on the left side of the y-axis. Now, unfold the paper and label the vertices of the image P", Q" and R".

Since the paper is folded along the y-axis, the line of reflection is y-axis.

Step 7 :

For the original figure, find the perpendicular distance from each point to the line of reflection.

From Point P to the line of reflection = 2 units

From Point Q to the line of reflection = 6 units

From Point R to the line of reflection = 2 units

Step 8 :

For the reflected figure, find the perpendicular distance from each point to the line of reflection.

From Point P" to the line of reflection = 2 units

From Point Q" to the line of reflection = 6 units

From Point R" to the line of reflection = 2 units

Step 9 :

From steps (7) and (8), it is clear that each point of the original figure PQR and its corresponding point on the image P"Q"R" must be at the same distance from the line of reflection.

Step 10 :

From steps (5) and (9), it is clear that the triangle PQR and its reflected objects across x-axis and y-axis satisfy the properties of reflection.

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