Question

Similar scalene triangle 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.

Answer 1

To determine the length of the third side of the second triangle using the SSS (Side-Side-Side) criterion and the properties of similar triangles, we start by setting up a proportion based on the given side lengths of the two triangles.

The first triangle has side lengths of \(1\), \(4\), and \(6\). Let's denote these as:
- \(a_1 = 1\)
- \(b_1 = 4\)
- \(c_1 = 6\)

The second triangle has corresponding sides \(3\) and \(12\). Denote these known lengths as:
- \(a_2 = 3\)
- \(b_2 = 12\)
- Let the length of the unknown third side be \(c_2\).

Since the triangles are similar, the ratios of their corresponding sides must be equal. Therefore, we have:

\[
\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}
\]

Calculating the ratios gives:

For the first known side:
\[
\frac{a_1}{a_2} = \frac{1}{3}
\]

For the second known side:
\[
\frac{b_1}{b_2} = \frac{4}{12} = \frac{1}{3}
\]

Since both ratios are equal, we can use them to find \(c_2\).

From the ratio of the corresponding sides:
\[
\frac{c_1}{c_2} = \frac{1}{3}
\]

Cross-multiplying gives:

\[
c_1 = \frac{1}{3} c_2
\]

Substituting \(c_1 = 6\) into the equation:

\[
6 = \frac{1}{3} c_2
\]

Multiplying both sides by 3 to isolate \(c_2\):

\[
c_2 = 6 \times 3 = 18
\]

Thus, the length of the third side of the corresponding triangle is \( \boxed{18} \).