Two triangles ΔABC and ΔDEF are similar. Their perimeters are 30 cm and 18 cm respectively. If side DE of triangle ΔDEF is 36 cm, use the perimeter–side ratio for similar triangles to find the corresponding side AB of triangle ΔABC (in centimeters) and choose the correct option.

Introduction / Context:
This question uses the property of similar triangles that the ratio of their corresponding sides is equal to the ratio of their perimeters. You are given the perimeters of two similar triangles ΔABC and ΔDEF and the length of a side DE in ΔDEF. The task is to find the corresponding side AB in ΔABC using proportional reasoning.

Given Data / Assumptions:

• Triangles ΔABC and ΔDEF are similar.
• Perimeter of ΔABC = 30 cm.
• Perimeter of ΔDEF = 18 cm.
• Side DE in ΔDEF corresponds to side AB in ΔABC and DE = 36 cm.
• All lengths are positive real numbers measured in centimeters.

Concept / Approach:
For similar triangles, the ratio of corresponding sides equals the ratio of their perimeters. Let k be the scale factor from ΔDEF to ΔABC. Then k = (perimeter of ΔABC) / (perimeter of ΔDEF). The corresponding side AB in ΔABC is then k times DE. We substitute the known values of the perimeters to compute k and then multiply by 36 cm to get AB.

Step-by-Step Solution:
Compute the ratio of perimeters: k = 30 / 18. Simplify the fraction: 30 / 18 = 5 / 3. Use similarity to relate corresponding sides: AB / DE = perimeter of ΔABC / perimeter of ΔDEF = 5 / 3. Therefore AB = DE * (5 / 3) = 36 * (5 / 3) = 12 * 5 = 60 cm.
Verification / Alternative check:
You can verify the proportionality by imagining that every side of ΔDEF is scaled by the same factor 5/3 to obtain ΔABC. Since DE is 36 cm, scaling by 5/3 yields 36 * 5/3 = 60 cm. The consistency of this factor would hold for all corresponding sides and the perimeter, confirming the calculation.

Why Other Options Are Wrong:
Option a ( 45 cm ) might come from using an incorrect ratio such as 30/24. Option b ( 50 cm ) and option d ( 40 cm ) correspond to arbitrary multipliers that do not match the perimeter ratio. Option e ( 72 cm ) could result from mistakenly using 18/30 instead of 30/18 and inverting the ratio. Only 60 cm fits the correct scale factor derived from the perimeters.

Common Pitfalls:
A frequent error is inverting the ratio of perimeters and thus scaling in the wrong direction. Another pitfall is confusing which triangle is larger from the perimeter values. Always match the numerator with the triangle whose side you are trying to find and the denominator with the triangle whose side you already know.

Final Answer:
The corresponding side AB of triangle ΔABC has length 60 cm.