The circumference of a triangle is 24 cm and the circumference of its in-circle is 44 cm. Then the area of the triangle is (taking `pi = (22)/(7)`)

Options

A. 56 square cm.
B. 84 square cm.
C. 48 square cm.
D. 68 square cm.

Answer

To find the area of the triangle given its circumference and the circumference of its in-circle, we can follow these steps:

### Step 1: Find the radius of the in-circle
The circumference of the in-circle is given as 44 cm. The formula for the circumference of a circle is:
\[ C = 2\pi R \]
where \( R \) is the radius of the circle. We can rearrange this formula to find \( R \):
\[ R = \frac{C}{2\pi} \]
Substituting the given circumference:
\[ R = \frac{44}{2 \times \frac{22}{7}} = \frac{44 \times 7}{2 \times 22} = \frac{44 \times 7}{44} = 7 \text{ cm} \]

### Step 2: Use the formula for the area of the triangle
The area \( A \) of a triangle can also be expressed in terms of its in-radius \( r \) and its semi-perimeter \( s \):
\[ A = r \times s \]
where \( s \) is the semi-perimeter of the triangle. The semi-perimeter is half of the perimeter (circumference) of the triangle.

### Step 3: Calculate the semi-perimeter
The circumference (perimeter) of the triangle is given as 24 cm. Therefore, the semi-perimeter \( s \) is:
\[ s = \frac{24}{2} = 12 \text{ cm} \]

### Step 4: Calculate the area of the triangle
Now we can substitute the values of \( r \) and \( s \) into the area formula:
\[ A = r \times s = 7 \times 12 = 84 \text{ cm}^2 \]

Thus, the area of the triangle is **84 square centimeters**.

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