Two triangles ΔABC and ΔPQR are similar. The area of ΔABC is 36 sq cm and the area of ΔPQR is 9 sq cm. If side PQ = 4 cm corresponds to side AB, what is the length of AB in centimetres?

Introduction:
This problem uses properties of similar triangles. When two triangles are similar, the ratio of their corresponding sides is related to the square root of the ratio of their areas. This question tests your understanding of this connection between side lengths and areas.

Given Data / Assumptions:

• Triangles ΔABC and ΔPQR are similar.
• Area of ΔABC = 36 sq cm.
• Area of ΔPQR = 9 sq cm.
• Side PQ corresponds to side AB.
• PQ = 4 cm.
• We must find AB in centimetres.

Concept / Approach:
For similar triangles, the ratio of their areas equals the square of the ratio of corresponding sides. If k is the ratio of a side of ΔABC to the corresponding side of ΔPQR, then: (Area of ABC) / (Area of PQR) = k^2. We first compute the area ratio, then find k, and finally compute AB using the given length of PQ.

Step-by-Step Solution:
Step 1: Compute the ratio of areas: Area(ABC) / Area(PQR) = 36 / 9 = 4. Step 2: Let k be the ratio AB / PQ. Then: k^2 = 4. Step 3: Take the positive square root (since lengths are positive): k = sqrt(4) = 2. Step 4: Now AB / PQ = 2, so AB = 2 * PQ. Step 5: Substitute PQ = 4 cm: AB = 2 * 4 = 8 cm.

Verification / Alternative check:
If AB = 8 cm and PQ = 4 cm, then the ratio of corresponding sides is 8 : 4 = 2 : 1. The ratio of areas should then be the square of this, that is 2^2 = 4. Indeed, 36 / 9 = 4, which matches, so the result is consistent with similarity rules.

Why Other Options Are Wrong:

• 16 and 12: These would give side ratios 16 : 4 = 4 and 12 : 4 = 3 respectively, leading to area ratios 16 and 9, which do not match 36 : 9.
• 6: This gives a side ratio of 6 : 4 = 1.5, area ratio 2.25, which is incorrect.
• 10: Also leads to an inconsistent area ratio when compared to 36 : 9.

Common Pitfalls:
Some learners directly compare side ratios with area ratios without squaring or taking square roots. Remember that area scales with the square of the linear scale factor in similar figures. Always relate side ratios and area ratios through the square relationship k^2.

Final Answer:
The length of AB is 8 cm.