The areas of two similar triangles ABC and DEF are `144 cm^(2) and 81 cm^(2)` respectively . If the longest side of the larger triangle is 36 cm , find the longest side of the other triangle .

Answer

To solve the problem, we need to use the properties of similar triangles. The areas of two similar triangles are proportional to the square of the ratio of their corresponding sides.

### Step-by-Step Solution:

1. **Identify the Areas of the Triangles**:
- Area of triangle ABC = 144 cm²
- Area of triangle DEF = 81 cm²

2. **Set Up the Ratio of the Areas**:
- The ratio of the areas of the two triangles can be expressed as:
\[ \frac{\text{Area of } ABC}{\text{Area of } DEF} = \frac{144}{81} \]

3. **Simplify the Ratio**:
- Simplifying \( \frac{144}{81} \):
\[ \frac{144 \div 9}{81 \div 9} = \frac{16}{9} \]

4. **Find the Ratio of the Sides**:
- Since the areas are proportional to the square of the sides, we take the square root of the area ratio to find the ratio of the corresponding sides:
\[ \frac{AB}{DE} = \sqrt{\frac{16}{9}} = \frac{4}{3} \]

5. **Use the Longest Side of Triangle ABC**:
- The longest side of triangle ABC is given as 36 cm. Let the longest side of triangle DEF be \( x \).
- According to the ratio of the sides:
\[ \frac{36}{x} = \frac{4}{3} \]

6. **Cross Multiply to Solve for \( x \)**:
- Cross multiplying gives:
\[ 4x = 36 \times 3 \]
\[ 4x = 108 \]
\[ x = \frac{108}{4} = 27 \text{ cm} \]

### Final Answer:
The longest side of triangle DEF is **27 cm**.