If the difference between the two sides of a right-angled triangle is 2 cm and the area of the triangle is `24 cm^(2)` , find the perimeter of the triangle.

Answer

To solve the problem step by step, let's follow the reasoning laid out in the video transcript.

### Step 1: Define the Variables
Let the two sides of the right-angled triangle be:
- \( A \) (one side)
- \( B \) (the other side)

According to the problem, we know:
- The difference between the two sides is 2 cm, which can be expressed as:
\[ A - B = 2 \]

### Step 2: Express One Variable in Terms of the Other
From the equation \( A - B = 2 \), we can express \( B \) in terms of \( A \):
\[ B = A - 2 \]

### Step 3: Use the Area Formula
The area \( A \) of a right-angled triangle is given by the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
In our case, substituting \( A \) and \( B \) into the area formula gives:
\[ \frac{1}{2} \times A \times B = 24 \]
Substituting \( B \) from Step 2:
\[ \frac{1}{2} \times A \times (A - 2) = 24 \]

### Step 4: Solve for A
Multiply both sides by 2 to eliminate the fraction:
\[ A \times (A - 2) = 48 \]
Expanding this gives:
\[ A^2 - 2A - 48 = 0 \]

### Step 5: Factor the Quadratic Equation
Now we need to factor the quadratic equation:
\[ A^2 - 2A - 48 = 0 \]
To factor, we look for two numbers that multiply to \(-48\) and add to \(-2\). The factors are:
\[ (A - 8)(A + 6) = 0 \]
Setting each factor to zero gives:
\[ A - 8 = 0 \quad \text{or} \quad A + 6 = 0 \]
Thus, \( A = 8 \) or \( A = -6 \). Since a side length cannot be negative, we take:
\[ A = 8 \text{ cm} \]

### Step 6: Find B
Now substitute \( A \) back into the equation for \( B \):
\[ B = A - 2 = 8 - 2 = 6 \text{ cm} \]

### Step 7: Find the Hypotenuse
Using the Pythagorean theorem, we can find the hypotenuse \( C \):
\[ C^2 = A^2 + B^2 \]
Substituting the values:
\[ C^2 = 8^2 + 6^2 = 64 + 36 = 100 \]
Taking the square root gives:
\[ C = \sqrt{100} = 10 \text{ cm} \]

### Step 8: Calculate the Perimeter
The perimeter \( P \) of the triangle is the sum of all sides:
\[ P = A + B + C = 8 + 6 + 10 = 24 \text{ cm} \]

### Final Answer
The perimeter of the triangle is:
\[ \boxed{24 \text{ cm}} \]