Let's carefully analyze the implications of dilating a rectangle by a scale factor of [tex]\( n = 1 \)[/tex].
1. Understanding Dilation with Scale Factor [tex]\( n = 1 \)[/tex]:
- Dilation is a transformation that alters the size of a geometric figure.
- The scale factor [tex]\( n \)[/tex] determines how much larger or smaller the image will be compared to the pre-image.
- A scale factor of [tex]\( n = 1 \)[/tex] means each dimension of the shape remains the same because:
- The length of each side of the rectangle is multiplied by 1: [tex]\( \text{new length} = \text{original length} \times 1 \)[/tex].
- This implies that the length and width of the rectangle do not change.
2. Evaluating Each Statement:
- Statement 1: "The image will be smaller than the pre-image because [tex]\( n = 1 \)[/tex]."
- Incorrect. Since the scale factor is 1, the image retains the original dimensions and does not get smaller.
- Statement 2: "The image will be congruent to the pre-image because [tex]\( n = 1 \)[/tex]."
- Correct. Congruent shapes are identical in form and size. With a scale factor of 1, the resulting image retains the exact dimensions of the pre-image, thus making it congruent.
- Statement 3: "The image will be larger than the pre-image because [tex]\( n = 1 \)[/tex]."
- Incorrect. A scale factor of 1 means there is no increase in dimensions, so the image size remains unchanged.
- Statement 4: "The image will be a triangle because [tex]\( n = 1 \)[/tex]."
- Incorrect. The shape remains a rectangle after dilation regardless of the scale factor, and changing a rectangle into a triangle is not a result of dilation.
3. Conclusion:
Given the analysis, the correct statement is:
- "The image will be congruent to the pre-image because [tex]\( n = 1 \)[/tex]."
Thus, the true statement is:
[tex]\[ \boxed{2} \][/tex]
