In the figure ABC is a triangle right angled at A . Find the area of the shaded region if AB = 6cm , BC = 10 cm and I is the centre of incircle triangle ABC .

Answer :

Here we have a right angle triangle ABC , Where $\angle$ A  =  90$°$ , And AB  =  6 cm And BC  =  10 cm And I is the center of incircle , we want to find the area of shaded portion of this triangle ABC .

Our figure is As :

Now we apply Pythagoras theorem in triangle ABC , and get

BC² =  AB² +  AC²

10² =  6² + AC²

AC² =  100 -  36

AC² =  64

AC  =  8 cm
And
We know Area of triangle = $\frac{1}{2}\times Base \times Height$ , So
Area of triangle ABC  = $\frac{1}{2}\times 6 \times 8 = 3 \times 8$ =  24 cm²
And
We know radius of incircle of right angle triangle  =  $\frac{mutiplication of two sides (\hspace{0.17em}Except hypotenuse )}{Sum of all three sides}$
So,
Radius of incircle with center " I " r  = $\frac{6 \times 8}{6 + 8 + 10} = \frac{48}{24}$ = 2 cm

We know area of circle  = $\mathrm{\pi }$r² , So

Area of incircle with center " I " = $\frac{22}{7}\times 2^2 = \frac{88}{7} = 12 .57 c{m}^2 ( As we know \pi = \frac{22}{7} )$
Then
Area of shaded portion  =  24  -  12.57 = 11.43 cm²                                ( Ans )