Question

A triangle has side lengths of 2, 6, and 7 units, and a similar but larger triangle has corresponding side lengths of 6, 18, and 21 units. Calculate and compare the perimeters of the triangles. What is the ratio of the corresponding sides of the smaller triangle to the larger triangle?(1 point)
Responses

3

1/9

1

1/3

Answer 1

To calculate the perimeters of the triangles, we first find the perimeter of each triangle.

**For the smaller triangle with side lengths 2, 6, and 7:**
\[
\text{Perimeter} = 2 + 6 + 7 = 15 \text{ units}
\]

**For the larger triangle with side lengths 6, 18, and 21:**
\[
\text{Perimeter} = 6 + 18 + 21 = 45 \text{ units}
\]

Now we can compare the perimeters:
- Smaller triangle perimeter: 15 units
- Larger triangle perimeter: 45 units

**Next, let's calculate the ratio of the corresponding sides of the smaller triangle to the larger triangle.**

The corresponding sides are:
- Smaller triangle side: 2 units → Larger triangle side: 6 units
- Smaller triangle side: 6 units → Larger triangle side: 18 units
- Smaller triangle side: 7 units → Larger triangle side: 21 units

The ratio for each pair of corresponding sides is:
\[
\frac{2}{6} = \frac{1}{3}
\]
\[
\frac{6}{18} = \frac{1}{3}
\]
\[
\frac{7}{21} = \frac{1}{3}
\]

In all cases, the ratio of the smaller triangle's side to the larger triangle's corresponding side is \( \frac{1}{3} \).

Thus, the answer for the ratio of the corresponding sides of the smaller triangle to the larger triangle is:
\[
\text{Ratio} = \frac{1}{3}
\]

**Final Answer: \(\frac{1}{3}\)**.