Question

The two triangles are similar. The smaller triangle has sides 5 ​inches, 10 ​inches, and 11 inches. The 11​-inch side on the smaller triangle corresponds to a side of 20 inches of the larger triangle. What is the perimeter of the larger​ triangle?

Answer 1

Since the two triangles are similar, the sides of the larger triangle will be proportional to the sides of the smaller triangle.

The ratio of the sides of the larger triangle to the smaller triangle can be determined using the corresponding sides. The 11-inch side of the smaller triangle corresponds to the 20-inch side of the larger triangle. The ratio \( k \) is calculated as follows:

\[
k = \frac{20 \text{ inches}}{11 \text{ inches}} = \frac{20}{11}
\]

Now, we will multiply each side of the smaller triangle by this scale factor \( k \) to find the lengths of the corresponding sides of the larger triangle.

The sides of the smaller triangle are:
- 5 inches
- 10 inches
- 11 inches

Calculating the corresponding sides of the larger triangle:

1. For the side corresponding to 5 inches:
\[
5 \times \frac{20}{11} = \frac{100}{11} \text{ inches}
\]

2. For the side corresponding to 10 inches:
\[
10 \times \frac{20}{11} = \frac{200}{11} \text{ inches}
\]

3. For the side corresponding to 11 inches (already given as 20 inches):
\[
11 \times \frac{20}{11} = 20 \text{ inches}
\]

Now we add the lengths of the sides of the larger triangle to find the perimeter:

\[
\text{Perimeter} = \frac{100}{11} + \frac{200}{11} + 20
\]

To add these fractions, we first convert 20 inches into a fraction with a denominator of 11:

\[
20 = \frac{20 \times 11}{11} = \frac{220}{11}
\]

Now adding all the sides together:

\[
\text{Perimeter} = \frac{100}{11} + \frac{200}{11} + \frac{220}{11} = \frac{100 + 200 + 220}{11} = \frac{520}{11} \text{ inches}
\]

Calculating this gives:

\[
\frac{520}{11} \approx 47.27 \text{ inches}
\]

Thus, the perimeter of the larger triangle is

\[
\text{Perimeter} \approx 47.27 \text{ inches}
\]