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

Question

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

Caroline Aucoin · Accepted Answer

#=pi/3# Chord has traveled #pi/2# to #(2pi)/3# radians or #pi/6#radians #=180/6#Degrees #=30^@# Circumference of the circle with area=#4pi# is also #4pi# So Length of the chord can be derived as follows : #360^@# correspond to #4pi# #pi/6# radians or #30^@#will correspond to #(4pi)/360# x30 #=pi/3#

Claire Akin · Answer

undefined The length of the chord can be calculated using the formula: $ \text{chord length} = 2 \times \text{radius} \times \sin\left(\frac{\theta}{2}\right) $, where $ \theta $ is the angle subtended by the chord at the center of the circle. Given that the area of the circle is $ 4\pi $, we can find the radius using the formula for the area of a circle: $ \text{Area} = \pi r^2 $. Once we have the radius, we can substitute it and the angle $ \theta = \frac{2\pi}{3} - \frac{\pi}{2} $ into the chord length formula to find the length of the chord. Solving these equations gives the length of the chord as $ 2\sqrt{3} $ units.