Triangles ABC and DEF are similar. If side AB is 10 cm and the corresponding side DE is 6 cm, what is the ratio of the perimeter of triangle ABC to the perimeter of triangle DEF?

Introduction / Context:
This question tests the basic property of similar triangles that the ratio of their perimeters is the same as the ratio of any pair of corresponding sides. It is a common and direct result used in many geometry and mensuration problems.

Given Data / Assumptions:
- Triangle ABC is similar to triangle DEF.
- Side AB in triangle ABC corresponds to side DE in triangle DEF.
- AB = 10 cm.
- DE = 6 cm.
- We are asked to find the ratio of Perimeter(ABC) to Perimeter(DEF).

Concept / Approach:
For two similar triangles, corresponding sides are in proportion, and all linear dimensions scale by the same factor. Therefore:
Perimeter(ABC) / Perimeter(DEF) = AB / DE = BC / EF = AC / DF.
This means the perimeter ratio can be found directly from any pair of corresponding sides. We just compute AB / DE and express it as a simplified ratio.

Step-by-Step Solution:
Step 1: Note that AB corresponds to DE and AB = 10 cm, DE = 6 cm. Step 2: Compute the ratio AB / DE = 10 / 6. Step 3: Simplify the fraction 10 / 6 by dividing numerator and denominator by 2: 10 / 6 = 5 / 3. Step 4: Because the triangles are similar, the perimeter ratio must equal this side ratio. Step 5: Therefore Perimeter(ABC) : Perimeter(DEF) = 5 : 3.

Verification / Alternative Check:
If we imagine that each side of triangle ABC is scaled down by a factor of 3 / 5, we would obtain triangle DEF. That means every linear measure, including the perimeter, is multiplied by 3 / 5 when going from ABC to DEF, so the reverse ratio from ABC to DEF is 5 : 3. This confirms that the ratio 5 : 3 is correct and consistent with the idea of similarity.

Why Other Options Are Wrong:
Option b, 3 : 5, is the inverse ratio and would apply if DE were 10 cm and AB were 6 cm, not the other way around.
Option c, 25 : 9, is the square of the linear ratio 5 : 3 and would apply to area ratios rather than perimeter ratios.
Option d, 9 : 25, is the inverse of the area ratio and does not match any direct linear measurement relationship between the triangles.

Common Pitfalls:
One common mistake is confusing perimeter ratios with area ratios and squaring the side ratio by habit. Another error is inverting the ratio by writing DE : AB instead of AB : DE. Carefully identifying which triangle is in the numerator and remembering that perimeters scale linearly while areas scale with the square of the scale factor helps to avoid these issues.

Final Answer:
The ratio of the perimeters Perimeter(ABC) : Perimeter(DEF) is 5:3.