Difficulty: Easy
Correct Answer: 8
Explanation:
Introduction / Context:
This question is about similar triangles and the relationship between their areas and corresponding side lengths. If two triangles are similar, their areas are proportional to the square of the ratio of their corresponding sides. This concept is widely tested in geometry and aptitude exams.
Given Data / Assumptions:
Concept / Approach:
For similar triangles, if k is the ratio of corresponding sides (larger/smaller), then:
Area ratio = k^2
Here, area(ΔXYZ) / area(ΔPQR) = 100 / 25 = 4. So k^2 = 4 and k = 2. That means each corresponding side in the larger triangle is twice the corresponding side in the smaller one.
Step-by-Step Solution:
Given area ratio = 100 / 25 = 4.
Let side scale factor (XYZ : PQR) be k.
Then k^2 = 4 ⇒ k = 2 (taking positive root, as lengths are positive).
Since PQ = 4 cm, and XY corresponds to PQ, we have XY = k * PQ.
Thus XY = 2 * 4 = 8 cm.
Verification / Alternative check:
If all sides scale by factor 2, area scales by factor 2² = 4. A triangle with area 25 sq cm, when scaled by factor 2 in all dimensions, has area 25 * 4 = 100 sq cm, consistent with the given areas. So side 4 cm in the smaller triangle corresponds to a side 8 cm in the larger one.
Why Other Options Are Wrong:
16 and 20 cm would correspond to scale factors 4 or 5, which would give area ratios 16 or 25, not 4. 14 cm is also not consistent with any simple square factor relationship between 25 and 100.
Common Pitfalls:
Many students mistakenly take the area ratio directly as the side ratio, forgetting to take the square root. Always remember: area ratio = (side ratio)², so you must take the square root of the area ratio to find the side ratio.
Final Answer:
The length of side XY is 8 cm.
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