To find the corresponding side of triangle $PQR$ to side $AB$ in triangle $ABC$, we'll use the fact that corresponding sides of similar triangles are proportional. However, since we are only given the perimeters and one side of triangle $ABC$, we can assume the triangles are similar and use the perimeter ratio to find the corresponding side.

Step 1: Find the ratio of the perimeters

Given:

• Perimeter of $\triangle ABC = 34$ cm
• Perimeter of $\triangle PQR = 42$ cm

The ratio of the perimeters is: $\text{Ratio} = \frac{\text{Perimeter of } \triangle ABC}{\text{Perimeter of } \triangle PQR} = \frac{34}{42} = \frac{17}{21}$

Step 2: Use the ratio to find the corresponding side

Let $PQ$ be the side of $\triangle PQR$ corresponding to side $AB$ of $\triangle ABC$. Since the triangles are similar, the ratio of corresponding sides is the same as the ratio of the perimeters.

$\frac{AB}{PQ} = \frac{17}{21}$ Substitute $AB = 12$ cm into the equation:

$\frac{12}{PQ} = \frac{17}{21}$

Step 3: Solve for $PQ$

$PQ = \frac{12 \times 21}{17} = \frac{252}{17} \approx 14.82 \text{ cm}$

So, the corresponding side $PQ$ is approximately 14.82 cm.

Would you like more details or have any questions? Here are some related questions:

1. What are similar triangles, and how do you identify them?
2. How do you calculate the area of similar triangles given the side lengths?
3. What is the importance of proportionality in similar triangles?
4. How do you find the length of another corresponding side in similar triangles?
5. How does the Pythagorean Theorem apply to similar triangles?
6. Can triangles be similar if only their perimeters are proportional?
7. What is the difference between congruent and similar triangles?
8. How can you determine the ratio of areas of two similar triangles?

Tip: Always check if the triangles are indeed similar before applying the proportionality of their sides.