Similar triangles – perimeters to sides:\nΔABC ~ ΔPQR and Perimeter(ABC) : Perimeter(PQR) = 5 : 9. If PQ = 45 cm, find AB (in cm).

Introduction / Context:
In similar triangles, all corresponding linear measures (sides, heights, perimeters) are in the same ratio. If perimeter ratio is 5:9, then any corresponding side ratio is also 5:9 in the same order (ABC to PQR).

Given Data / Assumptions:

• ΔABC ~ ΔPQR; Perimeter ratio ABC:PQR = 5:9.
• Correspondence: AB ↔ PQ (standard order).
• PQ = 45 cm.

Concept / Approach:
Side scale factor (ABC relative to PQR) is 5/9. Hence AB = (5/9)*PQ. Substitute PQ = 45 to get AB. Ensure the order of similarity is respected (ABC to PQR).

Step-by-Step Solution:

Verification / Alternative check:
Any other corresponding side would scale the same way; consistency across all sides and perimeters is guaranteed by similarity.

Why Other Options Are Wrong:
15 and 20 correspond to incorrect scale factors; 16 is unrelated; 25 uniquely follows 5/9 of 45.

Common Pitfalls:
Inverting the ratio (using 9/5) or mixing the triangle order.

Final Answer:
25