A circle has a chord that goes from #( 2 pi)/3 # to #(11 pi) / 12 # radians on the circle. If the area of the circle is #12 pi #, what is the length of the chord?

Question

A circle has a chord that goes from #( 2 pi)/3  # to #(11  pi) /  12  # radians on the circle.  If the area of the circle is #12   pi  #, what is the length of the chord?

Sophia Alfred · Accepted Answer

#=2.72# A chord that goes from #2pi/3#to #11pi/12# so it travels the distance #11pi/12-2pi/3=pi/4#; or #pi/4-:2pi=1/8# of the Circumference of the Circle Area of the Circle#=pir^2=12pi# or #r^2=12# or #r=sqrt12# or #r=3.46# Circumference of the circle#=2pir=2(pi)(3.46)=21.77# Therefore Length of the chord#=21.77/8=2.72#

Violet Aldrich · Answer

undefined To find the length of the chord, you can use the formula:

$$ \text{Chord length} = 2 \times \text{radius} \times \sin\left(\frac{\text{central angle}}{2}\right) $$

First, calculate the radius of the circle using the formula for the area of a circle:

$$ \text{Area} = \pi r^2 $$

$$ 12\pi = \pi r^2 $$

$$ r^2 = 12 $$

$$ r = \sqrt{12} = 2\sqrt{3} $$

Now, find the central angle:

$$ \text{Central angle} = \left(\frac{11\pi}{12}\right) - \left(\frac{2\pi}{3}\right) = \frac{11\pi}{12} - \frac{8\pi}{12} = \frac{3\pi}{12} = \frac{\pi}{4} $$

Now, plug the values into the formula for the chord length:

$$ \text{Chord length} = 2 \times 2\sqrt{3} \times \sin\left(\frac{\pi}{8}\right) $$

$$ \text{Chord length} = 4\sqrt{3} \times \sin\left(\frac{\pi}{8}\right) $$

$$ \text{Chord length} \approx 4\sqrt{3} \times 0.3827 $$

$$ \text{Chord length} \approx 1.5308 \times 4\sqrt{3} $$

$$ \text{Chord length} \approx 6.123 $$

So, the length of the chord is approximately $6.123$ units.