The areas of two similar triangles ΔXYZ and ΔPQR are 100 sq cm and 25 sq cm respectively. If side PQ = 4 cm, what is the length of side XY (in centimetres)?

Introduction / Context:
This question involves similar triangles and the relationship between their areas and the lengths of corresponding sides. When two triangles are similar, the ratios of corresponding side lengths are equal, and the ratio of their areas equals the square of the ratio of their corresponding sides. Here, we use the given areas to deduce the side ratio and then find the unknown side length in the larger triangle based on a known side in the smaller triangle.

Given Data / Assumptions:

• Triangles ΔXYZ and ΔPQR are similar.
• Area of ΔXYZ = 100 sq cm.
• Area of ΔPQR = 25 sq cm.
• PQ is a side of ΔPQR and PQ = 4 cm.
• XY is the corresponding side of ΔXYZ that we need to find.

Concept / Approach:
If two triangles are similar with a side ratio k = (side of larger triangle) / (side of smaller triangle), then the ratio of their areas is k^2. Here, the area ratio is 100 : 25. Once we determine k from the area ratio, we can find the corresponding side XY by multiplying the known side PQ by k. This method avoids any need to compute altitudes or other advanced properties; the similarity and area relationship suffice.

Step-by-Step Solution:
Area of ΔXYZ = 100 sq cm, area of ΔPQR = 25 sq cm.So area ratio (larger to smaller) = 100 : 25 = 4 : 1.If the linear scale factor between corresponding sides is k, then k^2 = 4.So k = √4 = 2 (taking the positive value for lengths).Side PQ in the smaller triangle is 4 cm, so side XY in the larger triangle is XY = k * PQ = 2 * 4 = 8 cm.

Verification / Alternative check:
We can verify by imagining the smaller triangle scaled up by a factor of 2 in all dimensions. If every side becomes twice as long, the area scales by a factor of 2^2 = 4. Starting from area 25 sq cm, the larger triangle would then have area 25 * 4 = 100 sq cm, which matches the given area. This confirms that the linear scale factor k must be 2 and that the corresponding side XY is correctly computed as 8 cm.

Why Other Options Are Wrong:
Option 16 cm would correspond to a scale factor of 4, which would produce an area ratio of 16, not 4. Option 14 and 10 cm do not produce any consistent area ratio when compared with PQ = 4 cm. Option 20 cm would imply k = 5 and an area ratio of 25, again inconsistent with the given ratio of 100 : 25. Only 8 cm fits the required scale factor derived from the areas.

Common Pitfalls:
Many students incorrectly assume that the side ratio equals the area ratio, using 100 : 25 = 4 and then multiplying 4 by 4 to get 16 cm, which is not correct. The side ratio is the square root of the area ratio, not the ratio itself. Another mistake is to invert the area ratio and take 25 : 100, leading to incorrect scale factors. Always remember that lengths scale as the square root of area ratios in similar figures.

Final Answer:
The length of side XY is 8 cm.