When we dilate a geometric figure, each point on the figure is moved away from or towards a fixed point (the center of dilation) by a certain multiplier known as the scale factor. The effect of this dilation on the figure's dimensions is dictated by the value of the scale factor \( n \).
In this scenario, the scale factor \( n \) is given to be 1. Here’s a detailed explanation regarding the implications of a scale factor of 1:
1. Understanding Dilation with Scale Factor 1:
- Dilation involves resizing the figure but retaining its shape.
- If the scale factor \( n = 1 \), every point on the figure remains the same distance from the center of dilation.
- Therefore, each dimension (length and width) of the rectangle remains unchanged after dilation.
2. Effect on the Geometry of the Figure:
- Since the dimensions remain unchanged, the new image produced will be identical in size and shape to the original pre-image.
- This means the image resulting from the dilation will be congruent to the pre-image. Congruency indicates that the figures have the same size and shape.
3. Analyzing the Given Statements:
- "The image will be smaller than the pre-image because \( n = 1 \)."
This statement is false because the image size does not change; it stays the same.
- "The image will be congruent to the pre-image because \( n = 1 \)."
This statement is true. Since dilation with \( n = 1 \) does not alter the size of the figure, the image remains congruent to the pre-image.
- "The image will be larger than the pre-image because \( n = 1 \)."
This statement is false because there is no increase in size when \( n = 1 \).
- "The image will be a triangle because \( n = 1 \)."
This statement is false as dilation affects the size and position of the figure but retains its shape. A rectangle remains a rectangle after dilation with any scale factor, including 1.
Given the analysis, the correct statement regarding the image of the dilation is:
"The image will be congruent to the pre-image because \( n = 1 \)."
Thus, the index of the true statement is 2.
