Question

[EF] is the diameter of a circle of center O and of radius R. G is a of this circle, distinct from E and F. Prove that GEF is a right triangle.​

Answer 1

Explanation:

Given : O is the centre of the circle with radius r. AB, CD and EF are the diameters of the circle. ∠OAF = ∠OCB = 60°.

To Find : What is the area of the shaded region?​

Solution:

∠OAF = 60°

OA = OF = Radius

=> ΔOAF is Equilateral Triangle

∠OCB = 60°

OC = OB Radius

Hence ΔOCB is Equilateral Triangle

∠AOF = 60° , ∠BOC = 60°

=> ∠COF = 180° - 60° - 60° = 60° as AC is straight Line

∠DOE = ∠COF ( vertically opposite angle )

∠DOE = 60°

ΔODE is also an equilateral Triangle

Each sector has 60 ° angle

Area of shaded region = (60/360)πr² - (√3/4) r²

= r² (π/6 - √3/4)

= (r²/6) (π - 3√3/2)

Area of 3 shaded regions

= 3 (r²/6) (π - 3√3/2)

= (r²/2) (π - 3√3/2)

(r²/2) (π - 3√3/2) is the correct answer