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 step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Understand the Area of a Triangle
The area \( A \) of a right-angled triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, the base is \( 4x \) cm and the height is \( (2x - 1) \) cm. We know the area is \( 30 \) cm².

### Step 2: Set Up the Equation
Using the area formula, we can set up the equation:
\[ 30 = \frac{1}{2} \times 4x \times (2x - 1) \]
Multiplying both sides by 2 to eliminate the fraction gives:
\[ 60 = 4x(2x - 1) \]

### Step 3: Expand and Rearrange the Equation
Now, we expand the right-hand side:
\[ 60 = 8x^2 - 4x \]
Rearranging this gives us:
\[ 8x^2 - 4x - 60 = 0 \]

### Step 4: Simplify the Quadratic Equation
We can simplify the equation by dividing all terms by 4:
\[ 2x^2 - x - 15 = 0 \]

### Step 5: Factor the Quadratic Equation
Next, we will factor the quadratic equation. We need two numbers that multiply to \( -30 \) (the product of \( 2 \) and \( -15 \)) and add to \( -1 \). The numbers \( -6 \) and \( 5 \) work:
\[ 2x^2 - 6x + 5x - 15 = 0 \]
Now, we can group the terms:
\[ 2x(x - 3) + 5(x - 3) = 0 \]
Factoring out \( (x - 3) \):
\[ (2x + 5)(x - 3) = 0 \]

### Step 6: Solve for \( x \)
Setting each factor to zero gives us:
1. \( 2x + 5 = 0 \) → \( x = -\frac{5}{2} \) (not valid since side lengths cannot be negative)
2. \( x - 3 = 0 \) → \( x = 3 \)

### Step 7: Calculate the Lengths of the Sides
Now that we have \( x = 3 \), we can find the lengths of the sides:
- The first side (base) is:
\[ 4x = 4 \times 3 = 12 \text{ cm} \]
- The second side (height) is:
\[ 2x - 1 = 2 \times 3 - 1 = 6 - 1 = 5 \text{ cm} \]

### Step 8: Calculate the Hypotenuse
Using the Pythagorean theorem to find the hypotenuse \( h \):
\[ h^2 = (4x)^2 + (2x - 1)^2 \]
Substituting the values:
\[ h^2 = 12^2 + 5^2 = 144 + 25 = 169 \]
Thus, the hypotenuse \( h \) is:
\[ h = \sqrt{169} = 13 \text{ cm} \]

### Final Result
The lengths of the sides of the triangle are:
- Base: \( 12 \) cm
- Height: \( 5 \) cm
- Hypotenuse: \( 13 \) cm