The area of triangle two sides of which are `18 cm` and `10 cm` and its perimeter is `42 cm` will be-

Options

A. `14sqrt(11)cm^(2)`
B. `21sqrt(11)cm^(2)`
C. `35sqrt(11)cm^(2)`
D. None of these

Answer

To find the area of the triangle with two sides measuring 18 cm and 10 cm, and a perimeter of 42 cm, we can follow these steps:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( a = 18 \, \text{cm} \)
- \( b = 10 \, \text{cm} \)
- Let \( c \) be the third side.

### Step 2: Calculate the third side
The perimeter of the triangle is given by:
\[ a + b + c = 42 \, \text{cm} \]
Substituting the values of \( a \) and \( b \):
\[ 18 + 10 + c = 42 \]
This simplifies to:
\[ 28 + c = 42 \]
Now, solving for \( c \):
\[ c = 42 - 28 = 14 \, \text{cm} \]

### Step 3: Calculate the semi-perimeter
The semi-perimeter \( s \) is given by:
\[ s = \frac{a + b + c}{2} = \frac{42}{2} = 21 \, \text{cm} \]

### Step 4: Use Heron's formula to find the area
Heron's formula states that the area \( A \) of a triangle can be calculated using:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values we have:
\[ A = \sqrt{21(21 - 18)(21 - 10)(21 - 14)} \]
Calculating each term:
\[ A = \sqrt{21 \times 3 \times 11 \times 7} \]

### Step 5: Calculate the product inside the square root
Calculating the product:
\[ 21 \times 3 = 63 \]
\[ 63 \times 11 = 693 \]
\[ 693 \times 7 = 4851 \]
So, we have:
\[ A = \sqrt{4851} \]

### Step 6: Simplify the square root
To simplify \( \sqrt{4851} \):
- Factor \( 4851 \) to find perfect squares.
- \( 4851 = 3^2 \times 7 \times 11^2 \)
- Thus, \( \sqrt{4851} = 11 \sqrt{39} \)

### Final Answer
The area of the triangle is:
\[ A = 21 \sqrt{11} \, \text{cm}^2 \]

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