A rectangle is dilated by a scale factor of [tex][tex]$n=1$[/tex][/tex]. Which statement is true regarding the image of the dilation?
A. The image will be smaller than the pre-image because [tex]$n=1$[/tex]. B. The image will be congruent to the pre-image because [tex]$n=1$[/tex]. C. The image will be larger than the pre-image because [tex][tex]$n=1$[/tex][/tex]. D. The image will be a triangle because [tex]$n=1$[/tex].
When a rectangle undergoes a dilation with a scale factor [tex]\( n \)[/tex], the dimensions of the image change based on the value of [tex]\( n \)[/tex].
Here is a detailed explanation of the impact of different scale factors:
1. If [tex]\( n > 1 \)[/tex]: The image will be larger than the pre-image. This is because the scale factor greater than 1 stretches the shape, increasing its dimensions proportionally.
2. If [tex]\( n < 1 \)[/tex]: The image will be smaller than the pre-image. A scale factor less than 1 compresses the shape, decreasing its dimensions proportionally.
3. If [tex]\( n = 1 \)[/tex]: The image will retain the same dimensions as the pre-image. This is because multiplying the dimensions by 1 leaves them unchanged.
Now, let's apply this understanding to the given problem:
The problem states that the rectangle is dilated by a scale factor of [tex]\( n = 1 \)[/tex].
1. The image will be smaller than the pre-image because [tex]\( n=1 \)[/tex]. – This cannot be true because a scale factor of 1 does not reduce the size.
2. The image will be congruent to the pre-image because [tex]\( n=1 \)[/tex]. – This statement is correct. A dilation with a scale factor of 1 means that the size and shape of the image remain exactly the same as the pre-image. Congruent figures are identical in shape and size.
3. The image will be larger than the pre-image because [tex]\( n=1 \)[/tex]. – This is incorrect because a scale factor of 1 does not enlarge the shape.
4. The image will be a triangle because [tex]\( n=1 \)[/tex]. – This is also incorrect because dilation does not change the shape type of the figure. A rectangle remains a rectangle after dilation.
Hence, the true statement is:
The image will be congruent to the pre-image because [tex]\( n=1 \)[/tex].
