Answer :
To determine the slope of the line segment [tex]\(\overline{X^{\prime} Y^{\prime}}\)[/tex] after a dilation by a scale factor of 1.3 with the origin as the center of dilation, we need to understand the effect of the dilation on the slope of the line.
1. Understanding dilation:
- Dilation is a transformation that produces an image that is the same shape as the original, but is resized by a scaling factor relative to a center point.
2. Effect of dilation on slope:
- When a figure is dilated from the origin by a scale factor, the coordinates of each point are multiplied by this scale factor.
- However, the slope of the line, which is a measure of its steepness and direction, remains unchanged. This is because the ratios of the vertical and horizontal distances between points on the line remain constant.
3. Given the slope [tex]\(m\)[/tex] of [tex]\(\overline{XY}\)[/tex]:
- Since dilation does not change the slope of a line (it only changes the length of the line segments but preserves their direction), the slope of [tex]\(\overline{X^{\prime} Y^{\prime}}\)[/tex] will be the same as the slope of [tex]\(\overline{XY}\)[/tex].
4. Answer analysis:
- A. [tex]\(1.3 \times m\)[/tex]: This suggests that the slope is scaled by the factor of the dilation, which is incorrect.
- B. [tex]\(1.3 \times 1\)[/tex]: This is irrelevant as it does not consider the given slope [tex]\(m\)[/tex].
- C. [tex]\(1.3 \times(m+1)\)[/tex]: This also incorrectly modifies the slope.
- D. [tex]\(m\)[/tex]: This correctly states that the slope remains unchanged after dilation.
Thus, the correct answer is:
D. [tex]\(m\)[/tex]
1. Understanding dilation:
- Dilation is a transformation that produces an image that is the same shape as the original, but is resized by a scaling factor relative to a center point.
2. Effect of dilation on slope:
- When a figure is dilated from the origin by a scale factor, the coordinates of each point are multiplied by this scale factor.
- However, the slope of the line, which is a measure of its steepness and direction, remains unchanged. This is because the ratios of the vertical and horizontal distances between points on the line remain constant.
3. Given the slope [tex]\(m\)[/tex] of [tex]\(\overline{XY}\)[/tex]:
- Since dilation does not change the slope of a line (it only changes the length of the line segments but preserves their direction), the slope of [tex]\(\overline{X^{\prime} Y^{\prime}}\)[/tex] will be the same as the slope of [tex]\(\overline{XY}\)[/tex].
4. Answer analysis:
- A. [tex]\(1.3 \times m\)[/tex]: This suggests that the slope is scaled by the factor of the dilation, which is incorrect.
- B. [tex]\(1.3 \times 1\)[/tex]: This is irrelevant as it does not consider the given slope [tex]\(m\)[/tex].
- C. [tex]\(1.3 \times(m+1)\)[/tex]: This also incorrectly modifies the slope.
- D. [tex]\(m\)[/tex]: This correctly states that the slope remains unchanged after dilation.
Thus, the correct answer is:
D. [tex]\(m\)[/tex]
