A circle has a chord that goes from #( pi)/2 # to #(13 pi) / 6 # radians on the circle. If the area of the circle is #5 pi #, what is the length of the chord?

1 Answer
Teddy
Jun 27, 2018

The length of the chord is #sqrt5#.

Explanation:

Let the center of the circle be point #O#. Let the point at #pi/2# be point #A# and let the point at #(13pi)/6# be point #B#. Since #(13pi)/6=2pi+pi/6#, point #B# can be thought of as located at an angle of #pi/6#. Therefore, #m/_AOB=pi/2-pi/6=pi/3#. Since points #A# and #B# are located on the circle, #OA=OB# because they are both radii. Since #OA=OB#, we know that #m/_OAB=m/_OBA# and since #m/_AOB=pi/3#, we know that #m/_OAB=m/_OBA=m/_AOB=pi/3# and that #OA=OB=AB#.

The area of a circle is given by #A=pir^2# so we know #5pi=pir^2# or that #r=sqrt5#. Since #OA=r=sqrt5# and chord #AB=OA#, #AB=sqrt5#.

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