Given that the perimeter of $\triangle ABC$ is 34 cm, and the perimeter of $\triangle PQR$ is 42 cm, and that side $PQ$ is 12 cm, we need to find the corresponding side in $\triangle ABC$ that corresponds to side $PQ$.

Step 1: Find the ratio of the perimeters

The ratio of the perimeters of the two triangles 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 $AB$ be the side of $\triangle ABC$ that corresponds to side $PQ$ of $\triangle PQR$. The ratio of the corresponding sides is the same as the ratio of the perimeters:

$\frac{AB}{PQ} = \frac{17}{21}$ Given $PQ = 12$ cm, we can solve for $AB$:

$AB = \frac{17}{21} \times 12 = \frac{204}{21} = 9.71 \text{ cm}$

So, the corresponding side $AB$ in $\triangle ABC$ is approximately 9.71 cm.

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

1. How do you verify that two triangles are similar?
2. What are the properties of similar triangles?
3. How do you calculate the other corresponding sides in similar triangles?
4. What is the significance of the perimeter ratio in similar triangles?
5. Can the area ratio of two similar triangles be calculated using the side lengths?
6. How do you determine corresponding angles in similar triangles?
7. What happens to the ratio of sides if the triangles are scaled by a factor?
8. How can you check if a given pair of triangles is not similar?

Tip: Always remember that the ratio of corresponding sides in similar triangles is equal to the ratio of their perimeters.