Answer:
Area of the shaded region = 87.5 cm²
Step-by-step explanation:
Area of the shaded region = Area of the sector - Area of the triangle
Area of the sector = [tex]\frac{\theta}{360}\times (2\pi r)[/tex]
Here, θ = Central angle subtended by the arc
r = radius of the circle
Area of the given sector = [tex]\frac{120}{360}\times (\pi r^{2})[/tex]
= 48π
= 150.796 cm²
From ΔABC,
Central angle BAC = 120°
Since, AD is an angle bisector of ∠BAC,
m∠BAD = m∠CAD = 60°
∠ADC = 90° [By theorem, line from the center to any chord is a perpendicular bisector of the chord in a circle)
Now, cos(60) = [tex]\frac{AD}{AC}[/tex]
[tex]\frac{1}{2}=\frac{AD}{12}[/tex]
AD = 6
Similarly, sin(60) = [tex]\frac{DC}{AC}[/tex]
[tex]\frac{\sqrt{3} }{2}=\frac{DC}{12}[/tex]
DC = [tex]6\sqrt{3}[/tex]
Since, BC = 2(DC) = [tex]12\sqrt{3}[/tex]
Area of ΔABC = [tex]\frac{1}{2}(\text{Height})(\text{Base})[/tex]
= [tex]\frac{1}{2}(AD)(BC)[/tex]
= [tex]\frac{1}{2}(6)(12\sqrt{3})[/tex]
= [tex]36\sqrt{3}[/tex] cm²
= 62.35 cm²
Now area of the shaded region = (150.796 - 62.35)
= 87.456
≈ 87.5 cm²
