Calculate the area of a triangle whose sides are 8 cm, 6 cm and 10 cm.
(i) `20 cm^(2)`, (ii) `25 cm^(2)`, (iii) `24 cm^(2)`, (iv) `16 cm^(2)`

Options

A. `20 cm^2`
B. `25 cm^2`
C. `24 cm^(2)`
D. `16 cm^2`

Answer

To calculate the area of a triangle with sides measuring 8 cm, 6 cm, and 10 cm, we will 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:
- \( a = 8 \) cm
- \( b = 6 \) cm
- \( c = 10 \) cm

Calculating the semi-perimeter:
\[ s = \frac{8 + 6 + 10}{2} = \frac{24}{2} = 12 \text{ cm} \]

### Step 2: Apply Heron's formula to find the area
Heron's formula for the area \( A \) of a triangle is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]

Substituting the values we have:
- \( s = 12 \) cm
- \( a = 8 \) cm
- \( b = 6 \) cm
- \( c = 10 \) cm

Calculating each term:
\[ s - a = 12 - 8 = 4 \]
\[ s - b = 12 - 6 = 6 \]
\[ s - c = 12 - 10 = 2 \]

Now substituting these values into Heron's formula:
\[ A = \sqrt{12 \times 4 \times 6 \times 2} \]

### Step 3: Simplify the expression
Calculating the product inside the square root:
\[ 12 \times 4 = 48 \]
\[ 48 \times 6 = 288 \]
\[ 288 \times 2 = 576 \]

Now, we take the square root:
\[ A = \sqrt{576} = 24 \text{ cm}^2 \]

### Conclusion
The area of the triangle is \( 24 \text{ cm}^2 \).

### Answer
The correct option is (iii) \( 24 \text{ cm}^2 \).

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