Question

a dilation maps $\triangle klm$ onto $\triangle klm$. find the missing values.\
$kl = 5$ in, $kl = 10$ in\
$lm = 3$ in, $lm = \underline{\quad}$ in\
$km = 6$ in, $km = \underline{\quad}$ in\
\
$kl = 5$ in, $kl = 10$ in\
$lm = 3$ in, $lm = \square$ in\
$km = 6$ in, $km = \square$ in\
(simplify your answers.)

Explanation:

Step1: Find the scale factor

The scale factor \( k \) of the dilation is the ratio of the length of a corresponding side in the image to the length in the pre - image. For side \( KL \) and \( K'L' \), \( k=\frac{K'L'}{KL}=\frac{10}{5} = 2 \).

Step2: Calculate \( L'M' \)

Since dilation preserves the ratio of lengths (it is a similarity transformation), the length of \( L'M' \) is the scale factor times the length of \( LM \). So \( L'M'=k\times LM \). Substituting \( k = 2 \) and \( LM = 3 \) in, we get \( L'M'=2\times3=6 \) inches.

Step3: Calculate \( K'M' \)

Similarly, the length of \( K'M' \) is the scale factor times the length of \( KM \). So \( K'M'=k\times KM \). Substituting \( k = 2 \) and \( KM = 6 \) in, we get \( K'M'=2\times6 = 12 \) inches.

Answer:

For \( L'M' \), the answer is \( 6 \) in. For \( K'M' \), the answer is \( 12 \) in.