Question

The area of a circle that is inscribed in a right angle triangle with sides of length $8cm,15cm$ and $17cm$ is

• A. $9\pi c{m}^2$
• B. $11\frac{1}{9}\pi c{m}^2$
• C. $14\pi c{m}^2$
• D. $10\pi c{m}^2$

Answer 1

The correct option is A $9\pi c{m}^2$

It is given that

A right angle triangle with sides of length $8cm,15cmand17cm$

Base = $8cm$

Height = $15cm$

Hypotenuse = $17cm$

To find the area of triangle

Area,$A=\frac{1}{2}\times Base\times height=\frac{1}{2}\times 8\times 15=60c{m}^2$

From the figure attached with this figure, triangle $ABC$ is a right triangle with a circle inscribed with radius $r$

To find radius $r$

Area of triangle $ABC=Ar\left(\Delta AOC\right)+Ar\left(\Delta AOB\right)+Ar\left(\Delta BOC\right)$

Here height of all triangle = $r$

Area, $A=(\frac{1}{2}\times AC\times r)+(\frac{1}{2}\times AB\times r)+(\frac{1}{2}\times BC\times r)=60$

$⟹\frac{1}{2}\times r[8+15+17]=60$

$⟹20r=60$

$⟹r=3cm$

To find area of circle

Area of circle = $\pi r^2=\pi \times 3\times 3=9\pi c{m}^2$