The ratio of the base and the height of a triangle is `3:2` and its area is `108 cm^(2)` .Find the length of its base and its height .

Answer

To solve the problem step by step, we will use the information given about the ratio of the base and height of the triangle and the area of the triangle.

### Step-by-Step Solution:

1. **Understand the Ratio**:
The ratio of the base to the height of the triangle is given as \(3:2\). This means if we let the base be \(3x\), then the height will be \(2x\).

**Hint**: Use variables to express the base and height based on the given ratio.

2. **Write the Area Formula**:
The area \(A\) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{Base} \times \text{Height} \]

3. **Substitute the Values**:
Substitute the expressions for base and height into the area formula:
\[ 108 = \frac{1}{2} \times (3x) \times (2x) \]

4. **Simplify the Equation**:
Simplifying the equation gives:
\[ 108 = \frac{1}{2} \times 6x^2 \]
\[ 108 = 3x^2 \]

5. **Solve for \(x^2\)**:
Divide both sides by 3:
\[ x^2 = \frac{108}{3} = 36 \]

6. **Find \(x\)**:
Take the square root of both sides:
\[ x = \sqrt{36} = 6 \text{ cm} \]

7. **Calculate the Base and Height**:
Now substitute \(x\) back to find the base and height:
- Base:
\[ \text{Base} = 3x = 3 \times 6 = 18 \text{ cm} \]
- Height:
\[ \text{Height} = 2x = 2 \times 6 = 12 \text{ cm} \]

8. **Final Answer**:
The length of the base is \(18 \text{ cm}\) and the height is \(12 \text{ cm}\).

### Summary of Results:
- Base = \(18 \text{ cm}\)
- Height = \(12 \text{ cm}\)