|TRIGONOMETRIC FUNCTIONS| What's the perimeter and area of the hatched region? Thank you!

The length of [AB] = 10 cm.
The angle α = 60°enter image source here

1 Answer
Jan 28, 2018

The area of the hatched region is #=34.24cm^2#. The perimeter is #=37.79cm#

Explanation:

Let #C# bethe third corner of the triangle

Then,

#BC=ABtan60=17.32#

The area of the triangle #Delta ABC# is

#A=1/2*AB*BC=1/2*10*17.32=86.6cm^2#

Let the intersection of the arc with the hypotenuse of the triangle be #=D#

The angle #alpha=60^@=pi/3#

Then,

The area of the sector #ADB# is

#A_1=1/2r^2alpha=1/2*AB^2*pi/3=1/2*10^2*pi/3=52.36cm^2#

The area of the shaded part is

#A-A_1=86.6-52.36=34.24cm^2#

The length of the arc #BD=r*alpha=10*pi/3=10.47cm#

The length of the hypotenuse is

#AC=(AB)/cosalpha=10/cos(60)=20cm#

The perimeter of the shaded region is

#P=BC+(AC-AD)+arcBD#

#=17.32+(20-10)+10.47#

#=37.79cm#