Triangles ABC and PQR are similar. The length of side AB is 18 centimetres and the length of the corresponding side PQ is 12 centimetres. If the area of triangle ABC is 324 square centimetres, find the area of triangle PQR.

Introduction / Context:
This question uses the concept of similar triangles and the relationship between the ratio of their corresponding sides and the ratio of their areas. It is a classic geometry topic that appears frequently in aptitude tests and school level examinations.

Given Data / Assumptions:

• Triangle ABC is similar to triangle PQR.
• Side AB in triangle ABC corresponds to side PQ in triangle PQR.
• AB = 18 cm.
• PQ = 12 cm.
• Area of triangle ABC = 324 square centimetres.
• We must find the area of triangle PQR.

Concept / Approach:
For two similar triangles:
Ratio of corresponding sides = k.
Ratio of their areas = k^2.
Here, k is the scale factor from one triangle to the other. We first find the linear ratio using AB and PQ, then square it to get the area ratio, and finally apply it to find the unknown area.

Step-by-Step Solution:
Step 1: Identify the corresponding sides: AB corresponds to PQ. Step 2: Compute side ratio for triangle PQR relative to ABC: k = PQ / AB = 12 / 18. Step 3: Simplify: 12 / 18 = 2 / 3. Step 4: Area ratio of PQR to ABC = k^2 = (2 / 3)^2 = 4 / 9. Step 5: Area of triangle ABC = 324 square centimetres. Step 6: Area of triangle PQR = (4 / 9) * 324. Step 7: Compute: 324 / 9 = 36, then 36 * 4 = 144 square centimetres.

Verification / Alternative check:
We can reverse the logic: If area of PQR is 144 square centimetres and area of ABC is 324 square centimetres, ratio is 144 / 324 = 4 / 9. Taking square root gives side ratio 2 / 3, which matches PQ / AB. This confirms our calculations and preserves the similarity relation correctly.

Why Other Options Are Wrong:
72 square centimetres: This corresponds to an area ratio of 2 / 9 rather than 4 / 9, which is too small.
108 square centimetres: Gives area ratio 108 / 324 = 1 / 3, not equal to k^2 where k is 2 / 3.
36 square centimetres: Far too small; would imply a much smaller side ratio inconsistent with 12 and 18.
487.5 square centimetres: Larger than the area of ABC, which contradicts the fact that PQ is smaller than AB.

Common Pitfalls:
Many students incorrectly assume that the area changes in the same ratio as the sides, forgetting to square the ratio. Another error is inverting the ratio and applying AB / PQ instead of PQ / AB for the target triangle. Always remember that if side ratio is a / b, then area ratio is (a / b)^2, and choose the direction of the ratio based on which triangle area you are trying to find.

Final Answer:
The area of triangle PQR is 144 square centimetres.