The lengths of three sides of a triangle are 20 cm, 16 cm and 12 cm. The area of the triangle is

Options

A. 96 `cm^(2)`
B. 120 `cm^(2)`
C. 144 `cm^(2)`
D. 160 `cm^(2)`

Answer

To find the area of a triangle with sides of lengths 20 cm, 16 cm, and 12 cm, we can use Heron's formula. Here’s how to calculate it step by step:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( a = 20 \) cm
- \( b = 16 \) cm
- \( c = 12 \) cm

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{20 + 16 + 12}{2} = \frac{48}{2} = 24 \text{ cm} \]

### Step 3: Apply Heron's formula
Heron's formula for the area \( A \) of the triangle is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Now, we will calculate each term:
- \( s - a = 24 - 20 = 4 \)
- \( s - b = 24 - 16 = 8 \)
- \( s - c = 24 - 12 = 12 \)

### Step 4: Substitute the values into Heron's formula
Now substituting the values into the formula:
\[ A = \sqrt{24 \times 4 \times 8 \times 12} \]

### Step 5: Calculate the product inside the square root
First, calculate the product:
\[ 24 \times 4 = 96 \]
\[ 96 \times 8 = 768 \]
\[ 768 \times 12 = 9216 \]

### Step 6: Take the square root
Now, we find the square root:
\[ A = \sqrt{9216} \]
Calculating the square root gives:
\[ A = 96 \text{ cm}^2 \]

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

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