Answer:
The area of the shaded part = 61.46
Step-by-step explanation:
Assume the hypotenuse of the triangle is c (c>0)
As the triangle inscribed in the semi circle is the right angle triangle, its hypotenuse is equal to the diameter of the circle.
The hypotenuse of the triangle can be calculated by Pythagoras theorem as following: [tex]c^{2} =a^{2} +b^{2} =(2\sqrt{5}) ^{2} + (4\sqrt{5}) ^{2} = 20 + 80 = 100[/tex]
=> c = 10
So that the semi circle has the diameter = 10 => its radius = 5
- The total area of 2 semi circles is equal to the area of the circle with radius =5
=> The total area of 2 semi circles is: [tex]\pi[/tex] x [tex]5^{2}[/tex] = 25[tex]\pi[/tex]
- The area of a triangle inscribed in the semi circle is: 1/2 x a x b = 1/2 x [tex]2\sqrt{5\\[/tex] x [tex]4\sqrt{5}[/tex] = 20
=> The area of 2 triangles inscribed in 2 semi circles is: 2 x 20 = 40
- The area of the square is: [tex]c^{2}[/tex] = [tex]10^{2} =100[/tex]
It can be seen that:
The area of the shaded part = The area of the square - The total area of 2 semi circles + The total are of 2 triangles inscribed in semi circles
= 100 - 25[tex]\pi[/tex] + 40 = 61.46
