Question : 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 $\pi =\frac{22}{7}$):
Option 1: 56 sq. cm
Option 2: 84 sq. cm
Option 3: 48 sq. cm
Option 4: 68 sq. cm
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Correct Answer: 84 sq. cm
Solution :
Given: AB + BC + CA = 24 cm
Let the incentre of the circle be O.
So, OD = OE = OF = radius
Now, $2\pi r = 44$
⇒ $2\times \frac{22}{7}\times r = 44$
⇒ $r=7$
$\text{Area of} \triangle ABC = \text{Area of} \triangle AOB +\text{Area of} \triangle AOC+\text{Area of}\triangle BOC$
$=\frac{1}{2}\times AB \times OF+\frac{1}{2}\times AC \times OE+\frac{1}{2}\times BC \times OD$
$=\frac{1}{2}\times OF(AB+BC+CA)$
$=\frac{1}{2}\times 7\times 24$ [$\because$ OD = OE = OF = radius = 7 cm]
$= 84$ sq. cm
Hence, the correct answer is 84 sq. cm.
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