Triangles ABC and DEF are similar. If the length of side AB is 18 cm and the length of the corresponding side DE is 10 cm, what is the ratio of the perimeter of triangle ABC to the perimeter of triangle DEF?

Introduction / Context:
This question tests the concept of similar triangles and the relationship between the ratios of corresponding sides and their perimeters. When two triangles are similar, the ratios of their corresponding sides are equal, and the ratio of their perimeters is the same as the ratio of any pair of corresponding sides. Understanding this principle allows us to find the ratio of the perimeters directly from a pair of corresponding sides.

Given Data / Assumptions:
- Triangle ABC is similar to triangle DEF.- Side AB in triangle ABC corresponds to side DE in triangle DEF.- AB = 18 cm.- DE = 10 cm.- We are asked for the ratio Perimeter(ABC) : Perimeter(DEF).

Concept / Approach:
For similar triangles, all corresponding side lengths are in the same ratio. If the ratio of any one pair of corresponding sides is known, that ratio is equal to the ratio of the perimeters, because perimeter is simply the sum of the three sides and each side is scaled by the same factor. Therefore, Perimeter(ABC) : Perimeter(DEF) = AB : DE. We simply reduce the fraction formed by these side lengths to its simplest form to get the required ratio.

Step-by-Step Solution:
Step 1: Recognise that ABC ~ DEF (triangles are similar).Step 2: Identify corresponding sides: AB corresponds to DE.Step 3: Given AB = 18 cm and DE = 10 cm.Step 4: The ratio of corresponding sides is AB : DE = 18 : 10.Step 5: Simplify 18 : 10 by dividing both numbers by 2: 18 / 2 = 9 and 10 / 2 = 5.Step 6: So, AB : DE = 9 : 5.Step 7: For similar triangles, the ratio of perimeters equals the ratio of corresponding sides.Step 8: Therefore, Perimeter(ABC) : Perimeter(DEF) = 9 : 5.

Verification / Alternative check:
Imagine triangle ABC has sides 18, 18k, and 18m for some positive constants k and m, and triangle DEF has sides 10, 10k, and 10m. Then Perimeter(ABC) = 18 + 18k + 18m = 18(1 + k + m). Perimeter(DEF) = 10(1 + k + m). The ratio of perimeters is 18(1 + k + m) : 10(1 + k + m). The common factor (1 + k + m) cancels, leaving 18 : 10, which reduces to 9 : 5. This reasoning confirms the direct method used earlier.

Why Other Options Are Wrong:
- 5 : 9: This inverts the correct ratio, treating the smaller triangle as having the larger perimeter.- 81 : 25: This is the square of the side ratio and corresponds to the ratio of areas, not perimeters.- 25 : 81: This inverts the area ratio and is not related to the perimeter ratio.- 3 : 2: Although 18 : 12 would simplify to 3 : 2, our sides are 18 and 10, not 18 and 12, so this ratio is not applicable.

Common Pitfalls:
A common mistake is to use the square of the side ratio when asked for the ratio of perimeters, confusing it with the ratio of areas. Another error is incorrectly identifying corresponding sides or inverting the ratio. Always remember that for perimeters, the ratio is the same as that of any pair of corresponding sides, while for areas, the ratio is the square of the side ratio. Keeping this distinction clear prevents such errors.

Final Answer:
The ratio of the perimeters of triangles ABC and DEF is 9 : 5.