Question

A similarity transformation consisting of a reflection

and a dilation is performed on a figure, and one point maps to itself. Explain how this can happen.

Answer 1

When a similarity transformation of reflection and dilation is performed on a figure and one point maps to itself, it means that the fixed point is the center of dilation and unaffected by the reflection.

Step-by-step explanation:

When a similarity transformation consisting of a reflection and a dilation is performed on a figure and one point maps to itself, it means that the figure undergoes a reflection and then a dilation with respect to that fixed point.

For example, let's say we have a triangle and we perform a reflection across a line and then dilate the figure with a scale factor of 2 from a fixed point. If one of the vertices of the triangle remains fixed during this transformation, that vertex will map to itself.

So, if a point is mapped to itself after a reflection and a dilation, it means that the point is the center of dilation and it is not affected by the reflection.

Answer 2

If the point in question is on the line of reflection, then the point won't move. Any point on a line of reflection stays where it is.

If that same point is the center of dilation, then it also doesn't move. The center of dilation is the only point that doesn't move in a dilation.

So the two conditions are:

• The point must be on the line of reflection
• The point must be the center of dilation

These two facts mean that the line of reflection must go through the center of dilation.