In a right angled triangle one of the perpendicular sides is `4` cm greater than the other and 4 cm lesser than the hypotensuse, Find the area of triangle in `cm^(2)`.

Options

A. 72
B. 48
C. 36
D. 96

Explanation

To solve the problem step by step, we need to define the sides of the right-angled triangle based on the information given.

### Step 1: Define the sides of the triangle
Let's denote the shorter perpendicular side as \( x \) cm. According to the problem:
- The other perpendicular side (the longer one) is \( x + 4 \) cm (4 cm greater than the shorter side).
- The hypotenuse is \( x + 4 + 4 = x + 8 \) cm (4 cm greater than the longer side).

### Step 2: Use the Pythagorean theorem
In a right-angled triangle, the relationship between the sides is given by the Pythagorean theorem:
\[ (\text{Hypotenuse})^2 = (\text{Base})^2 + (\text{Height})^2 \]
Substituting the values we defined:
\[ (x + 8)^2 = x^2 + (x + 4)^2 \]

### Step 3: Expand both sides
Now, we will expand both sides of the equation:
\[ (x + 8)^2 = x^2 + (x + 4)^2 \]
Expanding the left side:
\[ x^2 + 16x + 64 \]
Expanding the right side:
\[ x^2 + (x^2 + 8x + 16) = 2x^2 + 8x + 16 \]

### Step 4: Set up the equation
Now we can set the expanded forms equal to each other:
\[ x^2 + 16x + 64 = 2x^2 + 8x + 16 \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ 0 = 2x^2 + 8x + 16 - x^2 - 16x - 64 \]
This simplifies to:
\[ 0 = x^2 - 8x - 48 \]

### Step 6: Solve the quadratic equation
Now we can solve the quadratic equation \( x^2 - 8x - 48 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -8 \), and \( c = -48 \):
\[ x = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot (-48)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{8 \pm \sqrt{64 + 192}}{2} \]
\[ x = \frac{8 \pm \sqrt{256}}{2} \]
\[ x = \frac{8 \pm 16}{2} \]
This gives us two possible values for \( x \):
1. \( x = \frac{24}{2} = 12 \)
2. \( x = \frac{-8}{2} = -4 \) (not valid since length cannot be negative)

So, \( x = 12 \) cm.

### Step 7: Find the lengths of the sides
Now we can find the lengths of the sides:
- Shorter side: \( x = 12 \) cm
- Longer side: \( x + 4 = 12 + 4 = 16 \) cm
- Hypotenuse: \( x + 8 = 12 + 8 = 20 \) cm

### Step 8: Calculate the area of the triangle
The area \( A \) of a triangle is given by:
\[ A = \frac{1}{2} \times \text{Base} \times \text{Height} \]
Here, we can take the shorter side as the base and the longer side as the height:
\[ A = \frac{1}{2} \times 12 \times 16 = \frac{1}{2} \times 192 = 96 \text{ cm}^2 \]

### Final Answer
The area of the triangle is \( 96 \text{ cm}^2 \).