The sides of a triangle containing the right angle are 5x cm and `(3x-1)` cm. If the area of the triangle is 60 `cm^(2)`, calculate the lengths of the sides of the triangle.

Answer

To solve the problem step by step, we will follow the mathematical principles of area calculation for a triangle and solve for the unknown variable \( x \).

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
The two sides containing the right angle are given as \( 5x \) cm and \( (3x - 1) \) cm.

2. **Use the formula for the area of a triangle**:
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take \( 5x \) as the height and \( (3x - 1) \) as the base.

3. **Set up the equation using the given area**:
We know the area is \( 60 \, \text{cm}^2 \):
\[ 60 = \frac{1}{2} \times (3x - 1) \times 5x \]

4. **Multiply both sides by 2 to eliminate the fraction**:
\[ 120 = (3x - 1) \times 5x \]

5. **Expand the right-hand side**:
\[ 120 = 15x^2 - 5x \]

6. **Rearrange the equation to form a quadratic equation**:
\[ 15x^2 - 5x - 120 = 0 \]

7. **Divide the entire equation by 5 to simplify**:
\[ 3x^2 - x - 24 = 0 \]

8. **Factor the quadratic equation**:
We need to find two numbers that multiply to \( 3 \times -24 = -72 \) and add up to \( -1 \). The factors are \( -9 \) and \( 8 \):
\[ 3x^2 - 9x + 8x - 24 = 0 \]

9. **Group the terms**:
\[ (3x^2 - 9x) + (8x - 24) = 0 \]
Factor out the common terms:
\[ 3x(x - 3) + 8(x - 3) = 0 \]

10. **Factor out \( (x - 3) \)**:
\[ (3x + 8)(x - 3) = 0 \]

11. **Set each factor to zero**:
- From \( 3x + 8 = 0 \):
\[ x = -\frac{8}{3} \quad \text{(not valid since side lengths cannot be negative)} \]
- From \( x - 3 = 0 \):
\[ x = 3 \]

12. **Calculate the lengths of the sides**:
- For the height:
\[ 5x = 5 \times 3 = 15 \, \text{cm} \]
- For the base:
\[ 3x - 1 = 3 \times 3 - 1 = 9 - 1 = 8 \, \text{cm} \]

### Final Answer:
The lengths of the sides of the triangle are \( 15 \, \text{cm} \) and \( 8 \, \text{cm} \).

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