Question

if △abc ~ △def and the scale factor from △abc to △def is $\frac{1}{7}$, what are the lengths of $overline{de}$, $overline{ef}$, and $overline{df}$, respectively?
a. 21, 7, 28
b. 196, 98, 245
c. 4, 5, 2
d. 35, 21, 42

Explanation:

Step1: Recall similarity - side - length relationship

If $\triangle ABC\sim\triangle DEF$ with a scale factor of $k = \frac{1}{7}$, then if the side - lengths of $\triangle ABC$ are $AB$, $BC$, and $AC$, and the side - lengths of $\triangle DEF$ are $DE$, $EF$, and $DF$, we have $DE=\frac{1}{7}AB$, $EF = \frac{1}{7}BC$, and $DF=\frac{1}{7}AC$.

Step2: Identify side - lengths of $\triangle ABC$

Given $AB = 28$, $BC = 35$, and $AC = 14$.

Step3: Calculate side - lengths of $\triangle DEF$

$DE=\frac{1}{7}\times28 = 4$.
$EF=\frac{1}{7}\times35 = 5$.
$DF=\frac{1}{7}\times14 = 2$.

Answer:

C. 4, 5, 2