In similar triangles XYZ and PQR, the ratio of their perimeters is 3:2 and PQ = 6 cm; what is the length of side XY (in cm)?

Introduction / Context:
This question tests your knowledge of similar triangles and how their side lengths and perimeters are related. When two triangles are similar, all corresponding angles are equal and the ratios of their corresponding sides are equal. As a result, the ratio of their perimeters is also the same as the ratio of their corresponding sides. Understanding this relationship allows us to find missing side lengths without using trigonometry or complex formulas.

Given Data / Assumptions:

• Triangle XYZ is similar to triangle PQR.
• The ratio of their perimeters is given as perimeter(ΔXYZ) : perimeter(ΔPQR) = 3 : 2.
• Side PQ of triangle PQR is 6 cm.
• Side XY of triangle XYZ corresponds to PQ and we must find its length.

Concept / Approach:
For similar triangles, the scale factor between them is the same for all corresponding linear measures. This means:
(side of large triangle) / (corresponding side of small triangle) = (perimeter of large triangle) / (perimeter of small triangle) Here triangle XYZ is associated with the perimeter ratio 3, and triangle PQR with 2. Therefore, each corresponding side of XYZ is 3/2 times the length of the corresponding side of PQR.

Step-by-Step Solution:
Step 1: Write the perimeter ratio: perimeter(ΔXYZ) : perimeter(ΔPQR) = 3 : 2. Step 2: Conclude that the side ratio of corresponding sides is also 3 : 2. Step 3: Identify that side XY in ΔXYZ corresponds to side PQ in ΔPQR. Step 4: Let XY = k. Then k / 6 = 3 / 2 because corresponding sides follow the same ratio. Step 5: Cross-multiply to solve: 2k = 18. Step 6: Divide both sides by 2 to get k = 9. Step 7: Therefore, XY = 9 cm.

Verification / Alternative check:
We can view triangle PQR as the smaller triangle and triangle XYZ as a scaled-up version. The scale factor from PQR to XYZ is 3/2 = 1.5. If PQ is 6 cm, multiplying by 1.5 gives 9 cm. This matches our earlier result, confirming the calculation is consistent and logically sound.

Why Other Options Are Wrong:

• 4 cm and 8 cm: These are smaller than 9 cm and correspond to scale factors less than 3/2, which contradicts the given perimeter ratio.
• 12 cm: This would mean a scale factor of 2, which does not match the perimeter ratio of 3 : 2.
• 6 cm: This would suggest the triangles have the same size, whereas the ratio 3 : 2 clearly shows one triangle is larger.

Common Pitfalls:
A frequent mistake is inverting the ratio or assigning 3 to the smaller triangle and 2 to the larger one. Another error is assuming the side ratio is equal to the difference of perimeters instead of the ratio. Always align the ratio carefully with the given order and remember that all linear dimensions, including sides and perimeters, scale by the same factor in similar triangles.

Final Answer:
Thus, the length of side XY in triangle XYZ is 9 cm.