The base and the height of a triangle are in the ratio 5:3. If the area of the triangle is 67.5 `m^(2)`, find its base and height.

Answer

To solve the problem step by step, we will follow the given information about the triangle's base and height, and use the formula for the area of a triangle.

### Step-by-Step Solution:

1. **Understand the Ratio**:
The base and height of the triangle are in the ratio 5:3. We can express the base and height in terms of a variable \( x \).
- Let the base be \( 5x \) and the height be \( 3x \).

2. **Use the Area Formula**:
The area \( A \) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
We know the area is \( 67.5 \, m^2 \), so we can set up the equation:
\[ 67.5 = \frac{1}{2} \times (5x) \times (3x) \]

3. **Simplify the Equation**:
First, simplify the right side:
\[ 67.5 = \frac{1}{2} \times 15x^2 \]
This simplifies to:
\[ 67.5 = 7.5x^2 \]

4. **Eliminate the Fraction**:
To eliminate the fraction, multiply both sides of the equation by 2:
\[ 135 = 15x^2 \]

5. **Solve for \( x^2 \)**:
Now, divide both sides by 15:
\[ x^2 = \frac{135}{15} \]
Simplifying gives:
\[ x^2 = 9 \]

6. **Find \( x \)**:
Take the square root of both sides:
\[ x = 3 \]
(We ignore the negative value since lengths cannot be negative.)

7. **Calculate the Base and Height**:
Now that we have \( x \), we can find the base and height:
- Base: \( 5x = 5 \times 3 = 15 \, m \)
- Height: \( 3x = 3 \times 3 = 9 \, m \)

### Final Answer:
- Base = 15 meters
- Height = 9 meters