The area of a right angled triangle is 6 sq . Cm and its perimeter is 12 cm . The length of its hypotenuse is

Options

A. 6 cm
B. 5 cm
C. 2 cm
D. Data insufficient

Answer

To find the length of the hypotenuse of a right-angled triangle with an area of 6 sq. cm and a perimeter of 12 cm, we can follow these steps:

### Step 1: Set up the equations for area and perimeter
Let the lengths of the two legs of the triangle be \( x \) and \( y \). The area of the triangle is given by:
\[ \text{Area} = \frac{1}{2} \times x \times y = 6 \]
From this, we can derive:
\[ x \times y = 12 \quad \text{(1)} \]

The perimeter of the triangle is given by:
\[ \text{Perimeter} = x + y + \text{hypotenuse} = 12 \]
Let the hypotenuse be \( c \). Thus, we can write:
\[ x + y + c = 12 \quad \text{(2)} \]

### Step 2: Use the Pythagorean theorem
According to the Pythagorean theorem, we have:
\[ x^2 + y^2 = c^2 \quad \text{(3)} \]

### Step 3: Express \( c \) in terms of \( x \) and \( y \)
From equation (2), we can express \( c \):
\[ c = 12 - x - y \quad \text{(4)} \]

### Step 4: Substitute \( c \) into the Pythagorean theorem
Substituting equation (4) into equation (3):
\[ x^2 + y^2 = (12 - x - y)^2 \]

### Step 5: Expand the equation
Expanding the right-hand side:
\[ x^2 + y^2 = 144 - 24x - 24y + x^2 + y^2 + 2xy \]
Cancelling \( x^2 + y^2 \) from both sides gives:
\[ 0 = 144 - 24x - 24y + 2xy \]
Rearranging this, we get:
\[ 2xy - 24x - 24y + 144 = 0 \]

### Step 6: Substitute \( xy \) from equation (1)
From equation (1), we know \( xy = 12 \). Substituting this into the equation gives:
\[ 2(12) - 24x - 24y + 144 = 0 \]
This simplifies to:
\[ 24 - 24x - 24y + 144 = 0 \]
\[ -24x - 24y + 168 = 0 \]
Dividing the entire equation by -24:
\[ x + y = 7 \quad \text{(5)} \]

### Step 7: Solve for \( c \)
Now we can substitute equation (5) back into equation (4):
\[ c = 12 - (x + y) = 12 - 7 = 5 \, \text{cm} \]

### Conclusion
The length of the hypotenuse \( c \) is:
\[ \boxed{5 \, \text{cm}} \]