The areas of two similar triangles ΔABC and ΔPQR are 36 square centimetres and 9 square centimetres respectively. If side PQ = 4 cm, what is the length of side AB in centimetres?

Introduction / Context:
This question applies the properties of similar triangles to relate side lengths and areas. When two triangles are similar, the ratio of their areas is equal to the square of the ratio of any corresponding sides. Given the areas of two similar triangles and one side length in the smaller triangle, we can determine the corresponding side length in the larger triangle by using square roots of the area ratio.

Given Data / Assumptions:

• Triangles ABC and PQR are similar.
• Area of ΔABC = 36 sq cm.
• Area of ΔPQR = 9 sq cm.
• Side PQ = 4 cm, and PQ corresponds to side AB.
• We need the length of AB.

Concept / Approach:
For similar triangles, area ratio = (side ratio)^2. If we let k be the ratio of corresponding sides AB / PQ, then (Area of ABC) / (Area of PQR) = k^2. From this, we can solve for k by taking the square root of the area ratio. Once k is known, AB = k * PQ. The idea is to move from area ratio back to side ratio by square rooting.

Step-by-Step Solution:
Area of ΔABC = 36 sq cm.Area of ΔPQR = 9 sq cm.Area ratio = 36 / 9 = 4.If side ratio AB / PQ = k, then area ratio = k^2.So k^2 = 4, which means k = sqrt(4) = 2 (taking the positive value because lengths are positive).Given PQ = 4 cm, so AB = k * PQ = 2 * 4 = 8 cm.Therefore, AB = 8 cm.

Verification / Alternative check:
We can verify by checking that the area scale factor matches. If all linear dimensions of ΔABC are twice those of ΔPQR, then its area should be 2^2 = 4 times as large. Since area of ΔPQR is 9 sq cm, area of ΔABC should be 9 * 4 = 36 sq cm, which is exactly the given value. This confirms that the side length AB must indeed be 8 cm when PQ is 4 cm.

Why Other Options Are Wrong:
16 cm would correspond to a side ratio of 4, giving an area ratio of 16, not 4. 12 cm would give a side ratio of 3 and area ratio 9, inconsistent with the given 36 and 9. 6 cm implies a side ratio of 1.5 and area ratio 2.25, which does not match 4. 10 cm gives a non integer ratio that would not yield the given area ratio. Only 8 cm produces a side ratio of 2 and an area ratio of 4, consistent with the problem data.

Common Pitfalls:
A frequent mistake is to treat the area ratio directly as the side ratio, forgetting to take the square root. Another error is to invert the ratio of areas or to mix up which triangle is larger. Always identify which triangle has the larger area, compute the area ratio correctly, and then take the square root to get the scale factor for side lengths. Careful reading and methodical algebra prevent such errors.

Final Answer:
8