If an image of a triangle is congruent to the pre-image, what is the scale factor of the dilation?
A. 0.1
B. $\frac{1}{2}$
C. 1
D. 10
Answer (1)
Dilation is a transformation that changes the size of a figure by a scale factor.
Congruent figures have the same size and shape.
If the image is congruent to the pre-image after dilation, the size of the figure has not changed.
Therefore, the scale factor of the dilation is 1 .
Explanation
Problem Analysis Let's analyze the problem. We are given that a triangle is dilated, and the resulting image is congruent to the original triangle (pre-image). We need to find the scale factor of this dilation.
Dilation and Scale Factor Recall that dilation is a transformation that changes the size of a figure. The scale factor determines how much the figure is enlarged or reduced. If the scale factor is greater than 1, the figure is enlarged. If the scale factor is between 0 and 1, the figure is reduced. If the scale factor is exactly 1, the figure remains the same size.
Congruence and Scale Factor Since the image is congruent to the pre-image, the size and shape of the triangle have not changed. This means the dilation did not enlarge or reduce the triangle. Therefore, the scale factor must be 1.
Final Answer Thus, the scale factor of the dilation is 1.
Examples
Imagine you are using a photocopier to make a copy of a photograph. If you set the zoom to 100%, the copy will be exactly the same size as the original. This is analogous to a dilation with a scale factor of 1, where the image (copy) is congruent to the pre-image (original photograph). Similarly, in computer graphics, scaling an object by a factor of 1 preserves its original dimensions.
