Answer :
To determine the slope of the line segment [tex]\(\overline{X'Y'}\)[/tex] after dilation, let's carefully consider the impact of the dilation process.
1. Understand Dilation: Dilation is a transformation that changes the size of a figure but not its shape. When a line segment is dilated from an origin by a positive scale factor, the coordinate points defining the line segment are moved further from the origin by the scale factor.
2. Impact on Slope: The slope of a line segment is determined by the ratio of the vertical change to the horizontal change between two points on the line. Dilation affects the distances involved but not the ratio of these distances:
- If [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are endpoints of [tex]\(\overline{XY}\)[/tex] with slope [tex]\(m\)[/tex], dilation scales both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates proportionally.
- For example, if the coordinates are scaled by a factor of 1.3, the new coordinates will be [tex]\((1.3x_1, 1.3y_1)\)[/tex] and [tex]\((1.3x_2, 1.3y_2)\)[/tex].
- However, the slope is calculated as [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex]. After dilation, this becomes [tex]\(\frac{1.3y_2 - 1.3y_1}{1.3x_2 - 1.3x_1}\)[/tex], which simplifies to the original [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex].
3. Conclusion: Since the ratio of the vertical change to the horizontal change remains the same after dilation, the slope of the line segment does not change.
Thus, the slope of [tex]\(\overline{X'Y'}\)[/tex] is the same as the slope of [tex]\(\overline{XY}\)[/tex], which means the correct answer is:
[tex]\[ \boxed{D. \, m} \][/tex]
1. Understand Dilation: Dilation is a transformation that changes the size of a figure but not its shape. When a line segment is dilated from an origin by a positive scale factor, the coordinate points defining the line segment are moved further from the origin by the scale factor.
2. Impact on Slope: The slope of a line segment is determined by the ratio of the vertical change to the horizontal change between two points on the line. Dilation affects the distances involved but not the ratio of these distances:
- If [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are endpoints of [tex]\(\overline{XY}\)[/tex] with slope [tex]\(m\)[/tex], dilation scales both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates proportionally.
- For example, if the coordinates are scaled by a factor of 1.3, the new coordinates will be [tex]\((1.3x_1, 1.3y_1)\)[/tex] and [tex]\((1.3x_2, 1.3y_2)\)[/tex].
- However, the slope is calculated as [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex]. After dilation, this becomes [tex]\(\frac{1.3y_2 - 1.3y_1}{1.3x_2 - 1.3x_1}\)[/tex], which simplifies to the original [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex].
3. Conclusion: Since the ratio of the vertical change to the horizontal change remains the same after dilation, the slope of the line segment does not change.
Thus, the slope of [tex]\(\overline{X'Y'}\)[/tex] is the same as the slope of [tex]\(\overline{XY}\)[/tex], which means the correct answer is:
[tex]\[ \boxed{D. \, m} \][/tex]
