Answer :
To solve this problem, let's analyze what happens to a line segment when it's subject to dilation.
When a line segment [tex]\(\overline{XY}\)[/tex] is dilated from the origin by a scale factor, the coordinates of its endpoints change according to the scale factor. However, certain properties of the line segment remain unchanged under dilation. One of these properties is the slope.
1. Understanding the Slope:
- The slope of a line segment connecting two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex].
2. Effect of Dilation on Coordinates:
- When [tex]\(\overline{XY}\)[/tex] is dilated by a scale factor of 1.3 with the origin as the center of dilation, the coordinates of any point [tex]\((x, y)\)[/tex] on the line segment are transformed to [tex]\((1.3x, 1.3y)\)[/tex].
3. Effect of Dilation on the Slope:
- Consider the original points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] on [tex]\(\overline{XY}\)[/tex]. After dilation, these points become [tex]\((1.3x_1, 1.3y_1)\)[/tex] and [tex]\((1.3x_2, 1.3y_2)\)[/tex].
- We calculate the new slope using these transformed points:
[tex]\[ \text{slope of } \overline{X'Y'} = \frac{1.3y_2 - 1.3y_1}{1.3x_2 - 1.3x_1} = \frac{1.3(y_2 - y_1)}{1.3(x_2 - x_1)} = \frac{1.3}{1.3} \cdot \frac{y_2 - y_1}{x_2 - x_1} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
- Notice that the scale factor 1.3 cancels out in the numerator and the denominator, meaning the slope remains unchanged.
Thus, the slope of [tex]\(\overline{X'Y'}\)[/tex] after dilation is exactly the same as the slope of the original segment [tex]\(\overline{XY}\)[/tex], which is [tex]\(m\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{D. \, m} \][/tex]
When a line segment [tex]\(\overline{XY}\)[/tex] is dilated from the origin by a scale factor, the coordinates of its endpoints change according to the scale factor. However, certain properties of the line segment remain unchanged under dilation. One of these properties is the slope.
1. Understanding the Slope:
- The slope of a line segment connecting two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex].
2. Effect of Dilation on Coordinates:
- When [tex]\(\overline{XY}\)[/tex] is dilated by a scale factor of 1.3 with the origin as the center of dilation, the coordinates of any point [tex]\((x, y)\)[/tex] on the line segment are transformed to [tex]\((1.3x, 1.3y)\)[/tex].
3. Effect of Dilation on the Slope:
- Consider the original points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] on [tex]\(\overline{XY}\)[/tex]. After dilation, these points become [tex]\((1.3x_1, 1.3y_1)\)[/tex] and [tex]\((1.3x_2, 1.3y_2)\)[/tex].
- We calculate the new slope using these transformed points:
[tex]\[ \text{slope of } \overline{X'Y'} = \frac{1.3y_2 - 1.3y_1}{1.3x_2 - 1.3x_1} = \frac{1.3(y_2 - y_1)}{1.3(x_2 - x_1)} = \frac{1.3}{1.3} \cdot \frac{y_2 - y_1}{x_2 - x_1} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
- Notice that the scale factor 1.3 cancels out in the numerator and the denominator, meaning the slope remains unchanged.
Thus, the slope of [tex]\(\overline{X'Y'}\)[/tex] after dilation is exactly the same as the slope of the original segment [tex]\(\overline{XY}\)[/tex], which is [tex]\(m\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{D. \, m} \][/tex]
