Question

Similar scalene triangles are drawn on a coordinate plane. One triangle has side lengths of 1, 4, and 6. The other triangle has side lengths of 3 and 12, corresponding to the first triangle. Use the SSS criterion to determine the third side length of the corresponding triangle.(1 point)

Answer 1

To determine the third side length of the corresponding triangle, we can use the property of similar triangles. The ratios of the corresponding side lengths of similar triangles are equal.

Let's denote the side lengths of the first triangle as \( a = 1 \), \( b = 4 \), and \( c = 6 \). The second triangle has one side corresponding to \( a \), let’s denote it as \( a' = 3 \), and another side corresponding to \( b \), which is \( b' = 12 \). We need to find the length of the side \( c' \) that corresponds to \( c \).

Using the SSS (Side-Side-Side) similarity criterion, we can set up the following ratios based on the known corresponding side lengths:

\[
\frac{a'}{a} = \frac{b'}{b} = \frac{c'}{c}
\]

First, we can calculate the similarity ratio using the sides \( a \) and \( a' \):

\[
\frac{a'}{a} = \frac{3}{1} = 3
\]

Next, we can check this ratio using sides \( b \) and \( b' \):

\[
\frac{b'}{b} = \frac{12}{4} = 3
\]

Since both ratios are equal and correspond to the same factor, we have confirmed that these triangles are similar with a scale factor of 3.

Now we can apply this scale factor to determine \( c' \):

\[
c' = c \times 3 = 6 \times 3 = 18
\]

Thus, the third side length of the corresponding triangle is

\[
\boxed{18}.
\]