Question:
The perimeter of a triangle is 30 cm and the circumference of its incircle is 88 cm. The area of the triangle is
(a) $70 \mathrm{~cm}^{2}$
(b) $140 \mathrm{~cm}^{2}$
(c) $210 \mathrm{~cm}^{2}$
(d) $420 \mathrm{~cm}^{2}$
Solution:
We have to find the area of the given triangle.
Perimeter of triangle is 30 cm.
Let the radius of the circle be r.
We have,
Circumference of incircle $=88$
$2 \pi r=88$
$r=14$
Therefore,
$a r(\Delta \mathrm{ABC})=a r(\Delta \mathrm{OAB})+a r(\Delta \mathrm{OBC})+a r(\Delta \mathrm{OAC})$
$=\frac{1}{2}(r)(\mathrm{AB}+\mathrm{BC}+\mathrm{CA})$
$=\frac{1}{2}(14)(30) \mathrm{cm}^{2}$
$=210 \mathrm{~cm}^{2}$
Therefore the answer is (c).