Question

a triangle with side lengths 5, 12, and 13 is dilated to create a similar triangle with side lengths 30, 72, and 78. what is the dilation factor? a. 6 b. 1/2 c. 10 d. 5/6

Explanation:

Step1: Recall dilation - factor formula

The dilation factor $k$ for similar - triangles is found by dividing the length of a side of the dilated triangle by the length of the corresponding side of the original triangle.

Step2: Choose a pair of corresponding sides

Let's take the first pair of corresponding sides. If the original side length is $a_1 = 5$ and the dilated side length is $a_2=30$.
The dilation factor $k=\frac{a_2}{a_1}$.

Step3: Calculate the dilation factor

$k = \frac{30}{5}=6$.
We can check with other pairs. If we take the second - pair: original side length $b_1 = 12$ and dilated side length $b_2 = 72$, then $k=\frac{b_2}{b_1}=\frac{72}{12}=6$. For the third - pair: original side length $c_1 = 13$ and dilated side length $c_2 = 78$, then $k=\frac{c_2}{c_1}=\frac{78}{13}=6$.

Answer:

a. 6