If an image of a triangle is congruent to the pre-image, what is the scale factor of the dilation?
A. 0.1
B. [tex]$\frac{1}{2}$[/tex]
C. 1
D. 10
Answer (2)
Dilation is a transformation that changes the size of a figure.
Congruent figures have the same size and shape.
If the image is congruent to the pre-image after dilation, the size remains unchanged.
The scale factor 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 Definition Recall that dilation is a transformation that changes the size of a figure. If the scale factor is greater than 1, the figure gets larger. If the scale factor is between 0 and 1, the figure gets smaller. If the scale factor is exactly 1, the size of the figure remains unchanged.
Congruence Definition Also, remember that two figures are congruent if they have the same size and shape. In this case, the image and pre-image are congruent, meaning their sizes are the same.
Determining the Scale Factor Since the image is congruent to the pre-image, the size of the triangle does not change after the dilation. This means the scale factor must be 1, because multiplying the side lengths of the pre-image by 1 results in the same side lengths in the image.
Final Answer Therefore, the scale factor of the dilation is 1.
Examples
Imagine you have a photograph, and you want to create a copy that is exactly the same size as the original. If you use a photocopier to make a copy with a scale factor of 1, the copy will be congruent to the original photograph. This means the dimensions of the copy are identical to the dimensions of the original, preserving its size and shape. This is a practical example of dilation with a scale factor of 1, where the image remains unchanged.
The scale factor of the dilation is 1, indicating that the image is congruent to the pre-image. Therefore, the correct answer is C.
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