If the perimeter of a right-angled triangle is 56 cm and area of the triangle is 84 sq. cm, then the length of the hypotenuse is (in cm)

Options

A. 25
B. 50
C. 7
D. 24

Answer

To find the length of the hypotenuse \( C \) of a right-angled triangle given its perimeter and area, we can follow these steps:

### Step 1: Set up the equations
We know the following:
- The perimeter of the triangle is given by:
\[ A + B + C = 56 \quad \text{(1)} \]
- The area of the triangle is given by:
\[ \frac{1}{2} \times A \times B = 84 \quad \text{(2)} \]

### Step 2: Express \( A + B \) in terms of \( C \)
From equation (1), we can express \( A + B \) as:
\[ A + B = 56 - C \quad \text{(3)} \]

### Step 3: Express \( A \times B \) using the area
From equation (2), we can express \( A \times B \) as:
\[ A \times B = 2 \times 84 = 168 \quad \text{(4)} \]

### Step 4: Use the identity for squares
We know that:
\[ (A + B)^2 = A^2 + B^2 + 2AB \]
Substituting from equations (3) and (4):
\[ (56 - C)^2 = A^2 + B^2 + 2 \times 168 \]
This simplifies to:
\[ (56 - C)^2 = A^2 + B^2 + 336 \quad \text{(5)} \]

### Step 5: Use the Pythagorean theorem
For a right-angled triangle, we have:
\[ A^2 + B^2 = C^2 \quad \text{(6)} \]
Substituting equation (6) into equation (5):
\[ (56 - C)^2 = C^2 + 336 \]

### Step 6: Expand and rearrange the equation
Expanding the left side:
\[ 3136 - 112C + C^2 = C^2 + 336 \]
Now, cancel \( C^2 \) from both sides:
\[ 3136 - 112C = 336 \]

### Step 7: Solve for \( C \)
Rearranging gives:
\[ 3136 - 336 = 112C \]
\[ 2800 = 112C \]
Dividing both sides by 112:
\[ C = \frac{2800}{112} = 25 \]

### Conclusion
Thus, the length of the hypotenuse \( C \) is:
\[ \boxed{25 \text{ cm}} \]