Triangles ΔABC and ΔPQR are similar. The ratio of the perimeter of ΔABC to the perimeter of ΔPQR is 5 : 9. If side PQ = 45 cm, what is the length of side AB in centimetres?

Introduction / Context:
This question is about similar triangles and how ratios of their perimeters relate to ratios of corresponding sides. When two triangles are similar, every linear dimension of one triangle is in the same ratio to the corresponding dimension of the other triangle. The question gives you the ratio of perimeters and one side of the second triangle, and asks for the corresponding side of the first triangle.

Given Data / Assumptions:

• ΔABC is similar to ΔPQR.
• Perimeter ratio P(ABC) : P(PQR) = 5 : 9.
• Corresponding sides are AB ↔ PQ, BC ↔ QR, AC ↔ PR.
• PQ = 45 cm.
• We need to find AB.

Concept / Approach:
For similar triangles, the ratio of perimeters equals the ratio of corresponding side lengths. That means AB : PQ = 5 : 9. With PQ known, AB can be found by scaling 45 cm by the ratio 5/9. This avoids the need to know the actual perimeters and uses the similarity property directly and efficiently.

Step-by-Step Solution:
Given P(ABC) : P(PQR) = 5 : 9. For similar triangles, AB : PQ = 5 : 9. So AB / PQ = 5 / 9. PQ = 45 cm, so AB / 45 = 5 / 9. AB = 45 * (5 / 9) = 45 * 5 / 9. AB = 5 * 5 = 25 cm.

Verification / Alternative check:
The scale factor from ΔPQR to ΔABC for sides is 5/9. If PQ = 45 cm, the full perimeter of ΔPQR would be some multiple of 45, and the corresponding perimeter of ΔABC would be that multiple times 5/9. Although we do not need the full perimeters to answer the question, this reasoning is fully consistent with the side ratio used above, confirming that AB = 25 cm is correct.

Why Other Options Are Wrong:
15 cm and 20 cm: These correspond to smaller side ratios than 5/9 and would give perimeter ratios different from 5 : 9. 16 cm: Not a simple fraction of 45 in the ratio 5/9. 30 cm: This corresponds to a side ratio of 2/3 and would imply a perimeter ratio 2 : 3 rather than 5 : 9.

Common Pitfalls:
Mixing up the direction of the ratio and using 9/5 instead of 5/9. Thinking that perimeter ratio cannot be used directly for side ratios in similar triangles. Arithmetic mistakes in simplifying 45 * 5 / 9.

Final Answer:
The length of AB is 25 cm