The difference between the semi perimeter and the sides of a `Delta ABC` and 8 cm, 7 cm and 5 cm respectively. Find the area of the triangle.

Options

A. `40 sqrt(14) cm^(2)`
B. `30 sqrt(14) cm^(2)`
C. `10 sqrt(14) cm^(2)`
D. `20 sqrt(14) cm^(2)`

Explanation

To find the area of triangle ABC given the differences between the semi-perimeter and the sides, we can follow these steps:

### Step 1: Define the sides and the semi-perimeter
Let the sides of the triangle be:
- \( a = 8 \, \text{cm} \)
- \( b = 7 \, \text{cm} \)
- \( c = 5 \, \text{cm} \)

The semi-perimeter \( S \) of the triangle is given by:
\[ S = \frac{a + b + c}{2} \]

### Step 2: Calculate the semi-perimeter
Substituting the values of \( a \), \( b \), and \( c \):
\[ S = \frac{8 + 7 + 5}{2} = \frac{20}{2} = 10 \, \text{cm} \]

### Step 3: Calculate \( S - a \), \( S - b \), and \( S - c \)
Now, we can find the differences:
- \( S - a = 10 - 8 = 2 \, \text{cm} \)
- \( S - b = 10 - 7 = 3 \, \text{cm} \)
- \( S - c = 10 - 5 = 5 \, \text{cm} \)

### Step 4: Use Heron's formula to find the area
Heron's formula states that the area \( A \) of the triangle can be calculated as:
\[ A = \sqrt{S \cdot (S - a) \cdot (S - b) \cdot (S - c)} \]
Substituting the values we have:
\[ A = \sqrt{10 \cdot 2 \cdot 3 \cdot 5} \]

### Step 5: Simplify the expression
Calculating the product inside the square root:
\[ A = \sqrt{10 \cdot 2 \cdot 3 \cdot 5} = \sqrt{300} \]

### Step 6: Further simplify \( \sqrt{300} \)
We can simplify \( \sqrt{300} \):
\[ \sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100} \cdot \sqrt{3} = 10\sqrt{3} \]

### Step 7: Final area calculation
Thus, the area of triangle ABC is:
\[ A = 10\sqrt{3} \, \text{cm}^2 \]