In triangles ΔXYZ and ΔPQR, it is given that ΔXYZ is similar to ΔPQR and the ratio of their perimeters is 4 : 9. If side PQ of ΔPQR is 27 cm, what is the length of the corresponding side XY (in cm)?

Introduction / Context:
This question tests a basic property of similar triangles related to their perimeters and side lengths. When two triangles are similar, all their corresponding linear measurements such as sides, heights, and perimeters are in the same ratio. Here, the ratio of perimeters of triangles ΔXYZ and ΔPQR is given as 4 : 9, and a side of the larger triangle is known. We need to find the corresponding side of the smaller triangle using this similarity ratio.

Given Data / Assumptions:
- ΔXYZ is similar to ΔPQR. - Ratio of perimeters: Perimeter(ΔXYZ) : Perimeter(ΔPQR) = 4 : 9. - Side PQ of ΔPQR is 27 cm. - XY is the side in ΔXYZ corresponding to side PQ in ΔPQR.

Concept / Approach:
For similar triangles, the ratio of corresponding sides is equal to the ratio of perimeters, which is also equal to the scale factor between the triangles. If the ratio of perimeters is 4 : 9, then the ratio of any corresponding side in the smaller triangle to the side in the larger triangle is 4 : 9. Therefore, we can express XY in terms of PQ using this ratio. This is a direct proportionality question and does not require any angle or coordinate calculation.

Step-by-Step Solution:
Step 1: Recognise that ΔXYZ and ΔPQR are similar, so all corresponding linear measures are in the same ratio. Step 2: The ratio of perimeters is given as 4 : 9. This is also the ratio of any corresponding sides of the triangles. Step 3: Let XY correspond to PQ. Then XY : PQ = 4 : 9. Step 4: Substitute PQ = 27 cm in the ratio XY : 27 = 4 : 9. Step 5: Write this as XY / 27 = 4 / 9. Step 6: Solve for XY: XY = 27 * (4 / 9). Step 7: Compute 27 * (4 / 9) = (27 / 9) * 4 = 3 * 4 = 12 cm. Step 8: Therefore the length of side XY is 12 cm.

Verification / Alternative check:
We can interpret the ratio 4 : 9 as saying that the smaller triangle ΔXYZ is scaled up by a factor of 9 / 4 to obtain ΔPQR. In other words, every side in ΔPQR is 9 / 4 times the corresponding side in ΔXYZ. Therefore XY = PQ * (4 / 9). Substituting PQ = 27 again gives 27 * (4 / 9) = 12 cm, which matches the previous calculation, confirming the answer.

Why Other Options Are Wrong:
Option 9 cm would correspond to a ratio of 9 : 27 = 1 : 3, which is not the given ratio 4 : 9. Option 16 cm would give a side ratio 16 : 27, which does not simplify to 4 : 9. Option 15 cm would give 15 : 27 = 5 : 9, again not matching the required 4 : 9 ratio.

Common Pitfalls:
A common error is to invert the ratio accidentally and use 9 : 4 instead of 4 : 9 when relating the smaller triangle to the larger one. Another mistake is to confuse the perimeter ratio with an area ratio; remember that area ratios of similar triangles are the squares of the side ratios, but here the question explicitly provides the perimeter ratio, which is a linear ratio. Carefully matching corresponding triangles and corresponding sides is also important to avoid misassociation.

Final Answer:
The length of side XY is 12 cm.