Question

a rectangle is dilated by a scale factor of n = 1. which statement is true regarding the image of the dilation?
the image will be smaller than the pre - image because n = 1.
the image will be congruent to the pre - image because n = 1.
the image will be larger than the pre - image because n = 1.
the image will be a triangle because n = 1.

Explanation:

1. Recall the concept of dilation: Dilation is a transformation that changes the size of a figure. The scale factor \( n \) determines how the size changes.

• If \( n>1 \), the image is larger than the pre - image.
• If \( 0 < n<1 \), the image is smaller than the pre - image.
• If \( n = 1 \), the image has the same size as the pre - image (congruent), and the shape remains the same.

1. Analyze each option:

• Option 1: If \( n = 1 \), the image is not smaller. A scale factor less than 1 makes the image smaller. So this option is wrong.
• Option 2: When the scale factor \( n=1 \), the dilation does not change the size of the rectangle. So the image (the dilated rectangle) is congruent to the pre - image (the original rectangle). This option is correct.
• Option 3: A scale factor greater than 1 makes the image larger. Since \( n = 1 \), the image is not larger. So this option is wrong.
• Option 4: Dilation preserves the shape of the figure. A rectangle dilated with any scale factor (including \( n = 1 \)) will still be a rectangle, not a triangle. So this option is wrong.

Answer:

The image will be congruent to the pre - image because \( n = 1 \).