Difficulty: Easy
Correct Answer: 9
Explanation:
Introduction / Context:
This question tests understanding of similar triangles and the relationship between their perimeters and side lengths. For similar triangles, the ratio of their perimeters equals the ratio of any pair of corresponding sides. Given this ratio and one side in the smaller triangle, you can find the corresponding side in the larger triangle.
Given Data / Assumptions:
Concept / Approach:
For similar triangles, ratio of corresponding sides equals ratio of perimeters. If the perimeter ratio (larger : smaller) is 3 : 2, then each corresponding side in the larger triangle is (3/2) times the corresponding side in the smaller triangle:
XY / PQ = 3 / 2
Step-by-Step Solution:
Given perimeter(ΔXYZ) : perimeter(ΔPQR) = 3 : 2.
Thus side ratio XY : PQ = 3 : 2.
We know PQ = 6 cm.
So XY / 6 = 3 / 2.
Cross multiply: 2 * XY = 3 * 6 = 18.
Therefore XY = 18 / 2 = 9 cm.
Verification / Alternative check:
If every side of ΔPQR is multiplied by factor 3/2 to get ΔXYZ, then the perimeter also multiplies by 3/2, which matches the given perimeter ratio of 3 : 2. This consistency confirms that our side scaling factor and the resulting XY value are correct.
Why Other Options Are Wrong:
4 cm would imply a side smaller than PQ, contradicting a larger perimeter for ΔXYZ. 8 cm gives a side ratio of 4/3, which would lead to perimeter ratio 4/3, not 3/2. 12 cm would give a side ratio of 2, implying a perimeter ratio of 2 : 1, again inconsistent with the given 3 : 2.
Common Pitfalls:
One common error is to invert the ratio and use 2 : 3 instead of 3 : 2, which would give a smaller triangle instead of a larger one. Another pitfall is to confuse perimeter ratio with area ratio; area would scale with the square of the side ratio, not linearly.
Final Answer:
The length of side XY is 9 cm.
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