Question

With vertices A, B and C of a triangle ABC as centres, arcs are drawn with radii 2 cm each as shown in the figure. If AB = 6 cm, BC = 8 cm and AC = 10 cm, then find the area of the shaded region.​

Answer 1

Step-by-step explanation:

Given:-

a=BC=48cm

b=AC=52cm

c=AB=20cm

Semi-perimeter of △ABC=

2

a+b+c

=

2

52+48+20

=60cm

By using Heron's formula,

Area of triangle,

=

s(s−a)(s−b)(s−c)

=

60(60−48)(60−52)(60−20)

=480cm

2

Now,

Area of sectors =

360

πθ

r

2

=

360

πr

2

(θ

1

+θ

2

+θ

2

)

=

360

3.14×36

×(180)=56.52cm

2

Therefore,

Area of shaded region = Area of triangle − Area of sectors

⇒ Area of shaded region =480−56.52=423.48cm

2

Hope its help..

Answer 2

Given,

AB = 6cm

BC = 8cm

AC = 10cm

arc of radius = 2cm

To find,

Area of the region of triangle excluding the arc

Solution,

We can solve this mathematical problem by using the following mathematical process.

The method of finding the area of the region of the triangle, excluding the arc of radius 2cm, is as follows.

We know that,

Area of the sectors = $\frac{A}{360} * pie*r^2+\frac{B}{360}*pie*r^2+\frac{C}{360}*pie*r^2$  

                                = $\frac{3.14*4}{360} (A+B+C)$

                                = $\frac{3.14*4*180}{360}$

                                = 6.28 cm²

Since 6cm, 8cm, and 10 cm is a Pythagoras triplet, it is a right-angled triangle.

Thus, the area of the triangle = $\frac{1}{2} * 6*8$

                                           = 24 cm²

As a result,

The area of the shaded region = (24 - 6.28)cm²

                                                   = 17.72 cm²

Thus, the area of the shaded region is 17.72 cm².