The sides of a right angled triangle containing the right angle are 4x cm and (2x-1) cm. If the area of the triangle is `30cm^(2)` calculate the lengths of its sides.

Answer

To solve the problem, we need to find the lengths of the sides of a right-angled triangle where the two sides containing the right angle are given as \(4x\) cm and \((2x - 1)\) cm, and the area of the triangle is \(30 \, \text{cm}^2\).

### Step-by-Step Solution:

1. **Write the formula for the area of a triangle**:
The area \(A\) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
For our triangle, we can take \(4x\) as the base and \((2x - 1)\) as the height.

2. **Set up the equation for the area**:
Given that the area is \(30 \, \text{cm}^2\), we can write:
\[ 30 = \frac{1}{2} \times 4x \times (2x - 1) \]

3. **Simplify the equation**:
Multiply both sides by \(2\) to eliminate the fraction:
\[ 60 = 4x \times (2x - 1) \]
Expanding the right side:
\[ 60 = 8x^2 - 4x \]

4. **Rearrange the equation**:
Move all terms to one side to form a quadratic equation:
\[ 8x^2 - 4x - 60 = 0 \]

5. **Simplify the quadratic equation**:
Divide the entire equation by \(4\):
\[ 2x^2 - x - 15 = 0 \]

6. **Factor the quadratic equation**:
We need to factor \(2x^2 - x - 15\). We look for two numbers that multiply to \(2 \times -15 = -30\) and add to \(-1\). The numbers are \(-6\) and \(5\):
\[ 2x^2 - 6x + 5x - 15 = 0 \]
Grouping the terms:
\[ 2x(x - 3) + 5(x - 3) = 0 \]
Factoring out \((x - 3)\):
\[ (2x + 5)(x - 3) = 0 \]

7. **Solve for \(x\)**:
Set each factor to zero:
\[ 2x + 5 = 0 \quad \Rightarrow \quad x = -\frac{5}{2} \quad (\text{not valid since length cannot be negative}) \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]

8. **Calculate the lengths of the sides**:
Substitute \(x = 3\) back into the expressions for the sides:
- For \(AB = 2x - 1\):
\[ AB = 2(3) - 1 = 6 - 1 = 5 \, \text{cm} \]
- For \(BC = 4x\):
\[ BC = 4(3) = 12 \, \text{cm} \]

9. **Find the hypotenuse using Pythagoras' theorem**:
The hypotenuse \(AC\) can be calculated using:
\[ AC^2 = AB^2 + BC^2 \]
\[ AC^2 = 5^2 + 12^2 = 25 + 144 = 169 \]
\[ AC = \sqrt{169} = 13 \, \text{cm} \]

### Final Answer:
The lengths of the sides of the triangle are:
- \(AB = 5 \, \text{cm}\)
- \(BC = 12 \, \text{cm}\)
- \(AC = 13 \, \text{cm}\)