Question

line segment lm is dilated by a scale factor of \\( \frac { 1 } { 3 } \\) to form \\( \overline { l ^ { \prime } m ^ { \prime } } \\). point h, which lies on \\( \overline { l m } \\), is the center of dilation. if the slope of \\( \overline { l m } \\) is 6, which of the following is true about \\( \overline { l ^ { \prime } m ^ { \prime } } \\)?
a. line segment \\( l ^ { \prime } m ^ { \prime } \\) has a slope of 2, and passes through point h.
b. line segment \\( l ^ { \prime } m ^ { \prime } \\) has a slope of 2, and does not pass through point h.
c. line segment \\( l ^ { \prime } m ^ { \prime } \\) has a slope of 6 and does not pass through point h.
d. line segment \\( l ^ { \prime } m ^ { \prime } \\) has a slope of 6 and passes through point h.

Explanation:

Step1: Recall the property of dilation

When a line segment is dilated with a center of dilation on the line segment, the slope of the original line segment and the dilated line segment is the same. Also, since the center of dilation \(H\) lies on \(\overline{LM}\), the dilated line segment \(\overline{L'M'}\) will also pass through \(H\).
The slope of \(\overline{LM}\) is \(6\). After dilation, the slope of \(\overline{L'M'}\) remains \(6\) (because dilation is a similarity transformation and does not change the slope of a line when the center of dilation is on the line). And because \(H\) is the center of dilation (a fixed - point in the dilation process for points on the line containing \(H\)), \(\overline{L'M'}\) passes through \(H\).

Answer:

D. Line segment \(L'M'\) has a slope of \(6\) and passes through point \(H\)