To find the perimeters of both triangles, we first calculate the perimeter of the smaller triangle with side lengths of 2, 6, and 7 units.
Perimeter of the smaller triangle: \[ P_{\text{smaller}} = 2 + 6 + 7 = 15 \text{ units} \]
Now, we calculate the perimeter of the larger triangle with side lengths of 6, 18, and 21 units.
Perimeter of the larger triangle: \[ P_{\text{larger}} = 6 + 18 + 21 = 45 \text{ units} \]
Comparison of the perimeters:
- Smaller triangle perimeter = 15 units
- Larger triangle perimeter = 45 units
The ratio of the perimeters of the smaller triangle to the larger triangle is: \[ \text{Ratio of perimeters} = \frac{P_{\text{smaller}}}{P_{\text{larger}}} = \frac{15}{45} = \frac{1}{3} \]
Next, we find the ratio of the corresponding sides of the smaller triangle to the larger triangle. The corresponding sides are:
- For the first pair: \(2\) (smaller) to \(6\) (larger)
- For the second pair: \(6\) (smaller) to \(18\) (larger)
- For the third pair: \(7\) (smaller) to \(21\) (larger)
Calculating the ratio for one pair: \[ \text{For first pair: } \frac{2}{6} = \frac{1}{3} \]
Similarly for the other pairs: \[ \text{For second pair: } \frac{6}{18} = \frac{1}{3} \] \[ \text{For third pair: } \frac{7}{21} = \frac{1}{3} \]
Thus, the ratio of the corresponding sides of the smaller triangle to the larger triangle is consistently: \[ \frac{1}{3} \]
Therefore, the final answers are:
- Ratio of corresponding sides of the smaller triangle to the larger triangle: \( \frac{1}{3} \)
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