Question

a right angled triangle ABC, right angled at B, a semi- circle is drawn with AC as diameter ifAB=BC=7cm then find the area of the shaded region?
I

​

Answer 1

△ABC is a right angled triangle. So, by Pythagoras theorem

AC  

2

=AB  

2

+BC  

2

=7  

2

+7  

2

=98

∴AC=  

98

​  

=7  

2

​  

 cm

Arc A−B−C is a semicircle. ∴AC=7  

2

​  

 cm is the diameter of the semicircle.

Area of the shaded portion = Area of semi-circle (A−B−C)− Area of △ABC

=  

2

1

​  

×  

4

πd  

2

 

​  

−  

2

1

​  

×AB×BC

=  

2×7×4

22×98

​  

−  

2

7×7

​  

 

=14 cm  

2

Answer 2

Answer:

In △ABC,∠A=90

o

By Pythagoras theorem,

BC

2

=AB

2

+AC

2

3

2

+4

2

9+16=25

∴BC=5cm

A(shadded region)=A(△ ABC)+A (semicircle AB) + (semicircle AC) - A(semicircle BAC)

=

2

1

×3×4+

2

1

×π(

2

3

)

2

+

2

1

×π(

2

4

)

2

−

2

1

×π(

2

5

)

2

=6+

8

6π

+2π−

8

25π

=6+2π−2π

=6cm

2