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?

Answer 1

To find the perimeters of the two triangles, we first calculate the perimeter of each triangle by summing the lengths of their sides.

**For the smaller triangle with side lengths 2, 6, and 7 units:**

\[
\text{Perimeter}_{\text{small}} = 2 + 6 + 7 = 15 \text{ units}
\]

**For the larger triangle with side lengths 6, 18, and 21 units:**

\[
\text{Perimeter}_{\text{large}} = 6 + 18 + 21 = 45 \text{ units}
\]

Now, let's summarize the perimeters:
- Perimeter of the smaller triangle: 15 units
- Perimeter of the larger triangle: 45 units

Next, we will compare the perimeters:
- The perimeter of the larger triangle is three times that of the smaller triangle: \( \frac{45}{15} = 3 \).

Next, we can find the ratio of the corresponding sides of the triangles.

Let's take one pair of corresponding sides to calculate the ratio:
- Smaller triangle side: 2
- Larger triangle side: 6

The ratio of the smaller triangle side to the larger triangle side is:

\[
\text{Ratio} = \frac{\text{Smaller triangle side}}{\text{Larger triangle side}} = \frac{2}{6} = \frac{1}{3}
\]

We check the ratio for another pair of corresponding sides:
- Smaller triangle side: 6
- Larger triangle side: 18

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

Lastly, check the last pair:
- Smaller triangle side: 7
- Larger triangle side: 21

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

In all cases, the ratio of the corresponding sides of the smaller triangle to the larger triangle is:

\[
\frac{1}{3}
\]

Thus, we conclude:
- The perimeter of the smaller triangle is \( 15 \) units.
- The perimeter of the larger triangle is \( 45 \) units.
- The ratio of the corresponding sides of the smaller triangle to the larger triangle is \( \frac{1}{3} \).