The problem involves determining the perimeter of the larger triangle $P_2$ given that the two triangles are similar and their corresponding sides are proportional.

Step-by-step solution:

1. Understand the proportions of similar triangles: Since the triangles are similar, their sides are proportional. The small triangle has a side of $1 \, \text{in}$ corresponding to $5 \, \text{in}$ in the larger triangle. The scaling factor (ratio of sides) is: $\text{Scaling factor} = \frac{\text{Larger triangle's side}}{\text{Smaller triangle's side}} = \frac{5}{1} = 5.$

2. Relate the perimeters: For similar triangles, the ratio of their perimeters is the same as the ratio of their corresponding sides. Therefore: $\frac{P_2}{P_1} = \text{Scaling factor}.$

3. Substitute known values: The perimeter of the smaller triangle is given as $P_1 = 5 \, \text{in}$. Using the scaling factor $5$: $P_2 = P_1 \times \text{Scaling factor} = 5 \times 5 = 25 \, \text{in}.$

Final Answer:

$P_2 = 25 \, \text{inches}.$

Let me know if you’d like further clarification! 😊

Relative Questions:

1. What other properties remain the same between similar triangles (e.g., angles)?
2. How would the scaling factor change if one side of the smaller triangle doubled?
3. What if the perimeter of the larger triangle was given—how would you find the smaller triangle's perimeter?
4. Can the scaling factor between areas also be determined from the given data?
5. How does the proportionality rule extend to 3D objects like pyramids?

Tip:

When solving problems involving similar figures, always confirm whether you're working with ratios of lengths, areas, or volumes—they scale differently!