ABC is a right angled triangle in which ∠A = 90°. Find the area of the shaded region if AB=6 cm. . .

Questions : ABC is a right angled triangle in which ∠A = 90°. Find the area of the shaded region if AB=6 cm, BC=10 cm & I is the centre of the incircle of ∆ABC.

Doubt by Muskan

Solution : 

∠A = 90° (Given)
By Pythagoras Theorem 
AC = √(BC^2-AB^2)
AC = √(10^2-6^2)
AC=√(100-36)
AC=√64
AC =8 cm 

We, tangents drawn from an external point to a the circle are equal in length. 

AQ=AP = x
BP=BR=6-x
 (Can you Guess where could be point R?)
CQ=CR=8-x

Now
BC = 10 cm
BR+CR=10cm
6-x+8-x=10
14-2x=10
14-10=2x
4=2x
x=4/2
x=2
Now, We can prove that all the angles of quadrilateral AQIP are 90° and adjacent sides are equal. Hence, AQIP must be a square. 
All sides of square are equal. 
So, Radius of Circle = 2 cm

Area of shaded region
= Area of Triangle - area of circle
= 1/2 ×6×8-π(2)^2
= 24-(22/7)×4
=24-(88/7)
=(168-88)/7
=80/7
=11.428
=11.43 cm^2