When we dilate a geometric figure like a rectangle, the scale factor [tex]\( n \)[/tex] determines how the size of the figure changes. Let's analyze the given scale factor [tex]\( n = 1 \)[/tex] step by step to determine the true statement regarding the image of the dilation.
1. Understanding Dilation with Scale Factor [tex]\( n = 1 \)[/tex]:
- Dilation involves resizing the figure by multiplying all its dimensions (length and width) by the scale factor [tex]\( n \)[/tex].
- A scale factor [tex]\( n = 1 \)[/tex] means every dimension of the figure is multiplied by 1.
- Mathematically, if the original rectangle has dimensions (length [tex]\( l \)[/tex] and width [tex]\( w \)[/tex]), after dilation with scale factor [tex]\( n = 1 \)[/tex], the new dimensions will be:
[tex]\[
\text{New length} = l \times 1 = l
\][/tex]
[tex]\[
\text{New width} = w \times 1 = w
\][/tex]
2. Effect of [tex]\( n = 1 \)[/tex] on the Size of Image:
- Since multiplying the dimensions by 1 does not change their values, the size of the rectangle remains the same.
- Therefore, the image of the rectangle after dilation will be of the same size as the original rectangle.
3. Conclusion Regarding Congruence:
- When a geometric figure remains the same size after a transformation, it is congruent to itself.
- Congruence means that the original figure and the image are identical in shape and size.
4. Analyzing the Given Statements:
- The image will be smaller than the pre-image because [tex]\( n=1 \)[/tex]: This is incorrect because the image is not smaller; it remains the same.
- The image will be congruent to the pre-image because [tex]\( n=1 \)[/tex]: This is correct because the image and pre-image are identical in shape and size.
- The image will be larger than the pre-image because [tex]\( n=1 \)[/tex]: This is incorrect because the image does not increase in size.
- The image will be a triangle because [tex]\( n=1 \)[/tex]: This is incorrect because the shape of the rectangle does not change to a triangle through dilation.
From the given analysis, the correct statement is:
- The image will be congruent to the pre-image because [tex]\( n=1 \)[/tex].
Thus, the correct statement is:
[tex]\[ \boxed{\text{The image will be congruent to the pre-image because } n = 1.} \][/tex]
