The areas of similar triangles ABC and PQR are 36 sq cm and 9 sq cm respectively. If side PQ = 4 cm, what is the length of side AB (in cm)?

Introduction / Context:
This question uses the concept of similar triangles and the relationship between their areas and side lengths. When two triangles are similar, the ratio of their areas equals the square of the ratio of their corresponding sides. This allows us to find unknown side lengths from area information, which is a standard and important technique in geometry and aptitude exams.

Given Data / Assumptions:

• Triangles ABC and PQR are similar.
• Area of triangle ABC is 36 square centimetres.
• Area of triangle PQR is 9 square centimetres.
• Side PQ of triangle PQR measures 4 cm.
• We need to find the length of side AB in triangle ABC corresponding to PQ.

Concept / Approach:
If two triangles are similar with a side ratio k = (side of larger triangle) / (corresponding side of smaller triangle), then the ratio of their areas is k^2. Symbolically:
Area(large) / Area(small) = (side_large / side_small)^2 We know the areas of both triangles and one corresponding side (PQ). From the area ratio we can find k, then use k to calculate AB.

Step-by-Step Solution:
Step 1: Write the ratio of the areas: Area(ABC) : Area(PQR) = 36 : 9. Step 2: Simplify 36 : 9 to 4 : 1. Step 3: Let the ratio of corresponding sides (AB : PQ) be k : 1. Then area ratio is k^2 : 1. Step 4: Set k^2 equal to 4, since the area ratio is 4 : 1. Step 5: Take the positive square root: k = 2. Step 6: This means AB / PQ = 2, so AB = 2 * PQ. Step 7: Substitute PQ = 4 cm to get AB = 2 * 4 = 8 cm.

Verification / Alternative check:
We can check by using the side ratio directly. If the smaller triangle has side 4 cm and the larger triangle has side 8 cm, then the linear scale factor is 2. The area scale factor should be 2^2 = 4. Multiplying the smaller area (9 sq cm) by 4 gives 36 sq cm, which matches the area of the larger triangle. This confirms our calculation is correct.

Why Other Options Are Wrong:

• 16 and 12: These correspond to scale factors of 4 and 3 respectively, which would give area ratios of 16 : 1 or 9 : 1, not 4 : 1.
• 6: This would give a side ratio of 1.5, leading to an area ratio of 2.25, which contradicts the given ratio of 4.
• 4: This is simply the smaller side PQ and would imply equal triangles, which is not supported by the area values.

Common Pitfalls:
A common mistake is to assume the side ratio equals the area ratio instead of taking the square root. Another error is mixing up which triangle is larger. Always identify the larger area and take the square root of the area ratio to obtain the correct side ratio. Then multiply the known side by this factor to get the desired length.

Final Answer:
Therefore, the length of side AB is 8 cm.