Question

A circle has a chord that goes from `( pi)/3` to `(2 pi) / 3` radians on the circle. If the area of the circle is `4 pi`, what is the length of the chord?

Answer 1

Length of the chord is 1

Explanation:

Given:`" "/_A = (2pi)/3-pi/3 = pi/3`

Known: `/_C=/_B`

`/_A+/_B+/_C = pi" radians" -> 180^o`

Thus `color(blue)(/_C=/_B = 1/2( pi-pi/3) = pi/3" radians" ->60^o)`

Thus `Delta ABC ->" equilateral triangle"`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Given:`" area"=4pi`

But `" area"=pir^2`

Equate through area

`" "4pi="area"=pir^2`

Divide both sides by `pi`

`=>4=r^2`

`=> r=2`

Side `b=c=2`
'~~~~~~~~~~~~~~~~~~~~~~~~
Known: `cos(/_C)= a/2 -:r=a/(2r)`

`=>2rcos(/_C)=a`

`=>2(2)cos(pi/3)=a`

but `cos(pi/3)=1/2`

`=>2(2)(1/2)=a`

`=>"chord "= a=1`