Question

Triangle ABC is right angled at A.
Find the area of the shaded region if AB=6cm BC=10cm,O is the centre of the incircle of triangle ABC.

Answer 1

QUESTION:
Triangle ABC is right angled at A.
Find the area of the shaded region if AB=6cm BC=10cm,O is the centre of the incircle of triangle ABC.

SOLUTION:

ABC is a right angled triangle at A.
BC=10 cm and AB=6cm.
Let O be the centre and R be the radius of circle.

AB,BC,CA are tangents to the circle at P,M and N.
So, IP=IM=IN=R (radius of the circle)

In triangle ABC, Using Pythagoras Theorem:

$bc {}^{2} = ab {}^{2} + ac {}^{2}$
$(10) {}^{2} = (6) {}^{2} + ac {}^{2}$
$ac {}^{2} = 100 - 36 = 64$
$ac = \sqrt{64}$
$ac = 8cm$
Now, The area of triangle ABC

$= \frac{1}{2} \times 8 \times 6$
$= 24cm {}^{2}$
Area of ABC = Area of IAB +Area of IBC +Area of ICA

$24 = \frac{1}{2} \times r(ab) + \frac{1}{2} \times r(bc) + \frac{1}{2} \times r(ca)$
$24 = \frac{1}{2} \times r(ab + bc + ca)$
$24 = \frac{1}{2} \times r (6 + 8 + 10)$
$24 = 12r$
$r = 2cm$

Hence, The radius of circle is 2 cm.

Area of circle

$= \pi \: r {}^{2}$
$3.14 \times 2 \times 2 = 12.56cm {}^{2}$
Now , The area of shaded region
$= 24 - 12.56$
$= 11.44cm {}^{2}$
So, The area of shaded region is 11.44 cm^2

Answer 2

Answer:- Area of the shaded region is 11.44 square cm.