Area of the shaded region in the figure below (As):
As=Total area of the circle - area of the sector of 120º +area of the triangle
1)We calculate the total area of this circle
Area of a circle=πr²
r=radius=9 cm
Area of a circle=π(9 cm)²=81π cm²
2)We calculate the area of the sector of 120º
Area of the sector=(120º/360º)πr²=(1/3)π(9 cm)²=(81/3)π cm²
3) we calculate the area of the triangle
1) we calculate the base of this triangle by cousine theorem:
a²=b²+c²-2bc(cos Ф)
a²=(9 cm)²+(9 cm)²-2(9cm)(9 cm)cos 120º=243 cm
a=√243 cm
2)we calculate the height of this triangle by Pythagoras theorem:
a²=b²+c²
This triangle can be divided in two right triangles:
hypotenuse=9 cm
leg 1=√243/2 cm
leg 2=height
(9 cm)²=(√243 /2 cm)²+height²
height²=√(81 cm²-60.75 cm²)=20.25 cm
height=√20.25 cm=4.5 cm
Area of the triangle=(base x height) / 2=(√243 cm * 4.5 cm)/2=20.25√3 cm²
As=Total area of the circle - area of the sector of 120º +area of the triangle
As=81π cm²-81/3 π cm²+20.25√3 cm²=54π+20.25√3 cm²
Answer: c (54π+20.25√3) cm²