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Answer:
The area of the shaded region is {π/3 - √3/4 } × r².
Among the given options option (a) {π/3 - √3/4 } × r² is the correct answer.
Step-by-step explanation:
Given :
Radius of a circle = r
∆ABC is an equilateral triangle.
In an equilateral triangle all the three angles are 60° each.
∠A = ∠B = ∠C = 60°
∠BAC = ½ ∠BOC
[Angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle]
60° = ½ ×∠BOC
60° × 2 = ∠BOC
∠BOC = 120°
Angle at the centre of a circle, θ = 120°
Area of the segment , A = {πθ/360 - sin θ /2 cos θ/2 }r²
A = {120°π/360° - sin 120°/2 cos 120°/2 }× r²
A = {π/3 - sin 60°cos 60°} × r²
A = {π/3 - √3/2 × 1/2} × r²
A = {π/3 - √3/4 } × r²
Area of the segment = {π/3 - √3/4 } × r²
Hence, the area of the shaded region is {π/3 - √3/4 } × r².
HOPE THIS ANSWER WILL HELP YOU….
EXPLANATION.
ΔABC is an equilateral triangle.
As we know that,
In equilateral triangle every angle is 60°.
⇒ ∠A = 60°.
⇒ ∠BAC = 1/2∠BOC.
⇒ 60° = 1/2∠BOC.
⇒ ∠BOC = 120°.
As we know that,
Formula of :
Area of segment = (π x θ/360 - sinθ/2.cosθ/2).
Area of shaded region = Area of segment BC.
Put the values in the equation, we get.
⇒ (π x 120/360 - sin(120°)/2. cos(120°)/2) x r².
⇒ (π/3 - sin(60°).cos(60°)) x r².
⇒ (π/3 - √3/2 x 1/2) x r².
⇒ (π/3 - √3/4) x r².
Option [A] is correct answer.