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

19
Start Fraction 1 over 9 End Fraction

1
1

3
3

13

Answer 1

To solve the problem, we first calculate the perimeters of both triangles and then find the ratio of the corresponding sides.

**Step 1: Calculate the perimeter of the smaller triangle.**

The side lengths of the smaller triangle are 2, 6, and 7 units.

\[
\text{Perimeter of the smaller triangle} = 2 + 6 + 7 = 15 \text{ units}
\]

**Step 2: Calculate the perimeter of the larger triangle.**

The side lengths of the larger triangle are 6, 18, and 21 units.

\[
\text{Perimeter of the larger triangle} = 6 + 18 + 21 = 45 \text{ units}
\]

**Step 3: Compare the perimeters of the two triangles.**

The perimeter of the smaller triangle is 15 units, while the perimeter of the larger triangle is 45 units.

**Step 4: Calculate the ratio of corresponding sides of the smaller triangle to the larger triangle.**

The corresponding sides are:
- 2 (smaller) to 6 (larger)
- 6 (smaller) to 18 (larger)
- 7 (smaller) to 21 (larger)

Calculating the ratio for one pair of corresponding sides:

\[
\frac{2}{6} = \frac{1}{3}
\]

Calculating the ratio for another pair:

\[
\frac{6}{18} = \frac{1}{3}
\]

Calculating for the third pair:

\[
\frac{7}{21} = \frac{1}{3}
\]

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

**Final Answers:**
- Perimeter of smaller triangle: 15 units
- Perimeter of larger triangle: 45 units
- Ratio of corresponding sides of the smaller triangle to the larger triangle: \(\frac{1}{3}\)

Therefore, the correct response from the options you provided is:

**Start Fraction 1 over 3 End Fraction**