A circle has a chord that goes from #( 5 pi)/6 # to #(5 pi) / 4 # radians on the circle. If the area of the circle is #21 pi #, what is the length of the chord?

Question

A circle has a chord that goes from #( 5 pi)/6  # to #(5  pi) /  4  # radians on the circle.  If the area of the circle is #21   pi  #, what is the length of the chord?

Sadie Adams · Accepted Answer

#c ~~ 5.579 #

Given: #"Area" = 21pi = pir^2#

Therefore, #r^2 = 21#

Given: The angle between two the radii on either end of the chord, #theta = (5pi)/4 - (5pi)/6 = (5pi)/12#

Let c = the length of the chord; its length can found using a variant of the Law of Cosines:

#c = sqrt(r^2 + r^2 - 2(r)(r)cos(theta))#

#c = sqrt(2r^2 - 2(r^2)cos(theta))#

#c = sqrt(2r^2(1 - cos(theta)))#

#c = sqrt(42(1 -cos((5pi)/12)))#

#c ~~ 5.579 #

Caroline Aucoin · Answer

undefined To find the length of the chord, we first need to find the radius of the circle. We know that the area of a circle is given by the formula $A = \pi r^2$, where $A$ is the area and $r$ is the radius.

Given that the area of the circle is $21\pi$, we can set up the equation as follows:

$$21\pi = \pi r^2$$

Solving for $r$, we get:

$$r^2 = \frac{21\pi}{\pi}$$
$$r^2 = 21$$

Taking the square root of both sides:

$$r = \sqrt{21}$$

Now that we have the radius, we can use it to find the length of the chord. The length of the chord in a circle is given by the formula $2r\sin(\frac{\theta}{2})$, where $r$ is the radius and $\theta$ is the angle subtended by the chord at the center of the circle.

Given that the angle $\theta$ is from $\frac{5\pi}{6}$ to $\frac{5\pi}{4}$, we can find the difference between these two angles to get the measure of $\theta$:

$$\theta = \frac{5\pi}{4} - \frac{5\pi}{6}$$
$$\theta = \frac{\pi}{2}$$

Now, we can plug in the values into the formula:

$$2(\sqrt{21})\sin\left(\frac{\pi}{4}\right)$$

Using the fact that $\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, we get:

$$2(\sqrt{21})\left(\frac{\sqrt{2}}{2}\right)$$
$$= \sqrt{21} \times \sqrt{2}$$
$$= \sqrt{42}$$

Therefore, the length of the chord is $\sqrt{42}$ units.