Areas of two similar triangles are 36 `cm^(2)` and 100 `cm^(2)` . If the length of a side of the larger triangle is 20 cm find the length of the corresponding side of the smaller triangle.

Answer

To solve the problem step by step, we will use the properties of similar triangles and the relationship between their areas and corresponding sides.

### Step-by-Step Solution:

1. **Identify the Areas of the Triangles**:
- Let the area of the smaller triangle (ABC) be \( A_1 = 36 \, cm^2 \).
- Let the area of the larger triangle (DEF) be \( A_2 = 100 \, cm^2 \).

2. **Write the Ratio of the Areas**:
- The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore, we can write:
\[ \frac{A_1}{A_2} = \left(\frac{s_1}{s_2}\right)^2 \]
where \( s_1 \) is the length of a side of the smaller triangle and \( s_2 \) is the length of the corresponding side of the larger triangle.

3. **Substitute the Known Values**:
- Substitute the areas into the equation:
\[ \frac{36}{100} = \left(\frac{s_1}{20}\right)^2 \]

4. **Simplify the Ratio**:
- Simplifying \( \frac{36}{100} \):
\[ \frac{36}{100} = \frac{9}{25} \]
Therefore, we have:
\[ \frac{9}{25} = \left(\frac{s_1}{20}\right)^2 \]

5. **Take the Square Root**:
- Taking the square root of both sides gives:
\[ \sqrt{\frac{9}{25}} = \frac{s_1}{20} \]
This simplifies to:
\[ \frac{3}{5} = \frac{s_1}{20} \]

6. **Cross-Multiply to Solve for \( s_1 \)**:
- Cross-multiplying gives:
\[ 3 \cdot 20 = 5 \cdot s_1 \]
Thus:
\[ 60 = 5s_1 \]

7. **Solve for \( s_1 \)**:
- Dividing both sides by 5:
\[ s_1 = \frac{60}{5} = 12 \, cm \]

### Final Answer:
The length of the corresponding side of the smaller triangle is \( 12 \, cm \).