Triangle JKL was dilated using the rule D Subscript M, one-third. The image, triangle J'K'L', is the result of the dilation.

Question

Point M is the center of dilation. Triangle J K L is dilated to form smaller triangle J prime K prime L prime. The length of M L prime is 2.5.

What is L'L?

5 units
7.5 units
10 units
12.5 units

Answer 1

To determine the length \( L'L \), we can use the principle of dilation.

In a dilation centered at point \( M \) with a scale factor of \( \frac{1}{3} \), each point in the original triangle \( JKL \) is moved closer to point \( M \) by a factor of \( \frac{1}{3} \). This means that if \( L \) is the original point and \( L' \) is the image after dilation, the relationship between the distances from \( M \) to \( L \) and from \( M \) to \( L' \) can be established.

Given that \( ML' = 2.5 \), we can set up the following relationships based on the scale factor:

1. Let \( ML \) be the distance from point \( M \) to point \( L \).
2. The relationship due to dilation is \( ML' = \frac{1}{3} ML \).

Substituting the known value:

\[ 2.5 = \frac{1}{3} ML \]

To find \( ML \), multiply both sides by 3:

\[ ML = 2.5 \times 3 = 7.5 \]

Next, \( L'L \) is the distance from point \( L' \) to point \( L \). This can be calculated as follows:

\[ L'L = ML - ML' = 7.5 - 2.5 = 5 \]

Thus, the length \( L'L \) is:

\[ \boxed{5 \text{ units}} \]