The sides of a triangle are 4 cm, 5 cm and 6 cm. The area of the triangle is equal to

Options

A. `(15)/(4)`
B. `(15)/(4)sqrt(7)`
C. `(4)/(15)sqrt(7)`
D. `(4)/(15)`

Answer

To find the area of a triangle with sides measuring 4 cm, 5 cm, and 6 cm, we can use Heron's formula. Here’s a step-by-step solution:

### Step 1: Calculate the semi-perimeter (S)
The semi-perimeter \( S \) of a triangle is calculated using the formula:
\[ S = \frac{A + B + C}{2} \]
where \( A \), \( B \), and \( C \) are the lengths of the sides of the triangle.

For our triangle:
- \( A = 4 \) cm
- \( B = 5 \) cm
- \( C = 6 \) cm

Calculating \( S \):
\[ S = \frac{4 + 5 + 6}{2} = \frac{15}{2} \text{ cm} \]

### Step 2: Apply Heron's formula
Heron's formula for the area \( A \) of the triangle is given by:
\[ A = \sqrt{S \cdot (S - A) \cdot (S - B) \cdot (S - C)} \]

### Step 3: Calculate \( S - A \), \( S - B \), and \( S - C \)
Now we need to calculate:
- \( S - A = \frac{15}{2} - 4 = \frac{15}{2} - \frac{8}{2} = \frac{7}{2} \)
- \( S - B = \frac{15}{2} - 5 = \frac{15}{2} - \frac{10}{2} = \frac{5}{2} \)
- \( S - C = \frac{15}{2} - 6 = \frac{15}{2} - \frac{12}{2} = \frac{3}{2} \)

### Step 4: Substitute values into Heron's formula
Now substituting the values into the formula:
\[ A = \sqrt{\frac{15}{2} \cdot \frac{7}{2} \cdot \frac{5}{2} \cdot \frac{3}{2}} \]

### Step 5: Simplify the expression
Calculating the product inside the square root:
\[ A = \sqrt{\frac{15 \cdot 7 \cdot 5 \cdot 3}{16}} \]
Calculating \( 15 \cdot 7 \cdot 5 \cdot 3 \):
- \( 15 \cdot 7 = 105 \)
- \( 105 \cdot 5 = 525 \)
- \( 525 \cdot 3 = 1575 \)

So we have:
\[ A = \sqrt{\frac{1575}{16}} = \frac{\sqrt{1575}}{4} \]

### Step 6: Simplify \( \sqrt{1575} \)
Now we can simplify \( \sqrt{1575} \):
\[ 1575 = 25 \cdot 63 = 25 \cdot 9 \cdot 7 = 225 \cdot 7 \]
Thus,
\[ \sqrt{1575} = \sqrt{225 \cdot 7} = 15\sqrt{7} \]

### Step 7: Final area calculation
Substituting back into the area formula:
\[ A = \frac{15\sqrt{7}}{4} \]

### Conclusion
The area of the triangle is:
\[ \text{Area} = \frac{15\sqrt{7}}{4} \text{ cm}^2 \]