Difficulty: Medium
Correct Answer: 6.4 cm
Explanation:
Introduction / Context:
This question tests your understanding of similar triangles and the relationship between their perimeters and corresponding side lengths. In geometry and aptitude exams, problems like this frequently appear to check whether you can move comfortably between ratios of perimeters and ratios of sides. When two triangles are similar, all corresponding linear measurements such as side lengths, heights, medians, and perimeters are in the same ratio. Here, you are given the ratio of the perimeters of two similar triangles XYZ and PQR and one side length from triangle PQR. You are asked to find the length of the corresponding side XY in triangle XYZ using this similarity relationship in a clear and methodical way.
Given Data / Assumptions:
• Triangle XYZ is similar to triangle PQR (ΔXYZ ~ ΔPQR).
• Ratio of perimeter of triangle XYZ to perimeter of triangle PQR is 16 : 9.
• Side PQ in triangle PQR corresponds to side XY in triangle XYZ.
• Length of PQ = 3.6 cm.
• We assume both triangles are non degenerate and the similarity is correctly stated.
Concept / Approach:
For similar triangles, the ratio of corresponding sides is equal to the ratio of their perimeters. If the perimeter of triangle XYZ is to the perimeter of triangle PQR as 16 : 9, then any corresponding side in XYZ is to the corresponding side in PQR in the same ratio 16 : 9. Therefore, to find XY, we simply multiply PQ by the scale factor 16 / 9. This is a direct application of similarity and proportional reasoning, without needing to know individual perimeters or angles, which makes it a straightforward but important concept for aptitude exams.
Step-by-Step Solution:
Step 1: Let k be the linear scale factor from triangle PQR to triangle XYZ. Since perimeter(XYZ) : perimeter(PQR) = 16 : 9, the side ratio is also XY : PQ = 16 : 9.
Step 2: Express the relationship between corresponding sides: XY / PQ = 16 / 9.
Step 3: Substitute PQ = 3.6 cm into the proportion: XY / 3.6 = 16 / 9.
Step 4: Solve for XY by cross multiplication: XY = 3.6 * (16 / 9).
Step 5: Simplify the multiplication: 3.6 * 16 = 57.6, and 57.6 / 9 = 6.4.
Step 6: Therefore, XY = 6.4 cm.
Verification / Alternative check:
You can check the reasonableness of the answer by comparing sizes. Since the perimeter ratio 16 : 9 is greater than 1, triangle XYZ is larger than triangle PQR. Therefore, every corresponding side in XYZ should be longer than the corresponding side in PQR. Here, PQ is 3.6 cm and the computed XY is 6.4 cm, which is indeed larger. Also, the factor 16 / 9 is approximately 1.78, and 3.6 * 1.78 is close to 6.4, confirming that the arithmetic is consistent and the direction of scaling is correct.
Why Other Options Are Wrong:
Option 4.8 cm is obtained if someone mistakenly uses a smaller factor or reverses the ratio. It does not match the perimeter ratio 16 : 9 when applied correctly. Option 3.2 cm is even smaller than PQ, which would imply that triangle XYZ is smaller than triangle PQR, contradicting the given perimeter ratio where 16 is greater than 9. Option 8.6 cm is larger than necessary and does not correspond to any simple proportional calculation based on 3.6 and 16 : 9, so it is not consistent with similar triangle relationships.
Common Pitfalls:
A common mistake is to invert the ratio and use 9 : 16 instead of 16 : 9, which would lead to a smaller value for XY. Another typical error is to assume that perimeter ratios are unrelated to side ratios, or to add or subtract the numbers in the ratio instead of forming a proper fraction. Some candidates also misidentify which side in one triangle corresponds to which side in the other, but in this problem the statement clearly links PQ with XY. Carefully reading the direction of the ratio and correctly applying the proportion avoids these errors.
Final Answer:
The length of side XY in triangle XYZ is 6.4 cm.
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