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

Question

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

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Answer 1 · 2018-01-28T14:21:20

The area of the hatched region is $= 34.24 c m^{2}$ $=34.24cm^2$ . The perimeter is $= 37.79 c m$ $=37.79cm$

Let $C$ $C$ bethe third corner of the triangle

Then,

$B C = A B tan 60 = 17.32$ $BC=ABtan60=17.32$

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

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

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

$=37.79cm$