Question

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

Explanation:

Step1: Recall Dilation Properties

Dilation preserves the slope of a line segment (since it's a similarity transformation, parallel lines remain parallel, so slope is unchanged). Also, the center of dilation (point H) lies on the original segment \( \overline{LM} \), so it must lie on the dilated segment \( \overline{L'M'} \) (as dilation maps lines through the center to themselves).

Step2: Analyze Slope and Point

The slope of \( \overline{LM} \) is 6, so the slope of \( \overline{L'M'} \) (after dilation) is also 6 (dilation doesn't change slope). Since H is the center of dilation (on \( \overline{LM} \)), \( \overline{L'M'} \) passes through H. Now check options:

• Option A: Slope 6, passes through H (matches our analysis).
• Option B: Slope 2 (wrong, slope should be 6), eliminate.
• Option C: Slope 2 (wrong), eliminate.
• Option D: Slope 6 (correct slope), but says "does not pass through H" (wrong, since H is center of dilation), eliminate.

Answer:

A. Line segment \( \boldsymbol{L'M'} \) has a slope of 6 and passes through point H.