To determine the effect of dilating a rectangle by a scale factor of [tex]\( n = 1 \)[/tex], let’s understand the concept of dilation. Dilation is a transformation that alters the size of a figure but not its shape. The scale factor, [tex]\( n \)[/tex], determines how much the figure is enlarged or reduced.
1. Dilation with [tex]\( n = 1 \)[/tex]:
- When the scale factor [tex]\( n \)[/tex] is 1, it means that every point of the original figure (pre-image) stays at the same distance from the center of dilation as it was before.
- Specifically, the coordinates of each point of the figure remain unchanged since multiplying any value by 1 results in the original value.
- Mathematically, if the original rectangle has vertices [tex]\((x_1, y_1)\)[/tex], [tex]\((x_2, y_2)\)[/tex], [tex]\((x_3, y_3)\)[/tex], and [tex]\((x_4, y_4)\)[/tex], then after dilation with [tex]\( n = 1 \)[/tex], the vertices of the dilated rectangle will still be [tex]\((x_1, y_1)\)[/tex], [tex]\((x_2, y_2)\)[/tex], [tex]\((x_3, y_3)\)[/tex], and [tex]\((x_4, y_4)\)[/tex].
2. Analyzing the given statements:
- The image will be smaller than the pre-image because [tex]\( n = 1 \)[/tex]:
- This is incorrect. A scale factor of 1 does not reduce the size of the figure; it keeps the size the same.
- The image will be congruent to the pre-image because [tex]\( n = 1 \)[/tex]:
- This is correct. Since all points of the figure remain unchanged and the distances between these points do not alter, the figure maintains the same size and shape. Therefore, the dilated image is congruent to the original pre-image.
- The image will be larger than the pre-image because [tex]\( n = 1 \)[/tex]:
- This is incorrect. A scale factor of 1 does not enlarge the figure, it keeps the size the same.
- The image will be a triangle because [tex]\( n = 1 \)[/tex]:
- This is incorrect. The dilation does not change the type of figure; it only changes the size. A rectangle will remain a rectangle.
3. Conclusion:
- The correct statement regarding the image of the dilation is: "The image will be congruent to the pre-image because [tex]\( n = 1 \)[/tex]."
