To find the perimeter of each triangle, we first calculate the sum of the side lengths for both triangles.
For the smaller triangle with side lengths of 2, 6, and 7: \[ \text{Perimeter of the smaller triangle} = 2 + 6 + 7 = 15 \text{ units} \]
For the larger triangle with side lengths of 6, 18, and 21: \[ \text{Perimeter of the larger triangle} = 6 + 18 + 21 = 45 \text{ units} \]
Now we can compare the perimeters:
- Perimeter of the smaller triangle: 15 units
- Perimeter of the larger triangle: 45 units
Next, we need to find the ratio of the corresponding sides of the smaller triangle to the larger triangle. We choose the corresponding sides:
- For the side of length 2 in the smaller triangle, the corresponding side in the larger triangle is 6.
- For the side of length 6 in the smaller triangle, the corresponding side in the larger triangle is 18.
- For the side of length 7 in the smaller triangle, the corresponding side in the larger triangle is 21.
The ratios of the corresponding sides are:
- \( \frac{2}{6} = \frac{1}{3} \)
- \( \frac{6}{18} = \frac{1}{3} \)
- \( \frac{7}{21} = \frac{1}{3} \)
Since all ratios of the corresponding sides are equal to \( \frac{1}{3} \), we can conclude that the ratio of the corresponding sides of the smaller triangle to the larger triangle is:
\[ \frac{1}{3} \]
So the final results are:
- The perimeter of the smaller triangle is 15 units, and the perimeter of the larger triangle is 45 units.
- The ratio of the corresponding sides of the smaller triangle to the larger triangle is \( \frac{1}{3} \).
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