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:

When a figure is dilated, if the scale factor \(n = 1\), the transformation rule is \((x,y)\to(nx,ny)=(x,y)\). Congruent figures have the same shape and size. Since the scale factor \(n = 1\) means the image has the same dimensions as the pre - image, the image and pre - image are congruent.

• Option 1: If \(n<1\) (e.g., \(n = 0.5\)), the image is smaller. But \(n = 1\) does not make the image smaller.
• Option 3: If \(n>1\) (e.g., \(n = 2\)), the image is larger. But \(n = 1\) does not make the image larger.
• Option 4: Dilation of a rectangle with any non - zero scale factor \(n\) results in a rectangle (not a triangle) as the shape of the figure (rectangle) is preserved in dilation.

Answer:

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