Question

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

The length of [AB] = 10 cm.
The angle α = 60°

Answer 1

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`