A circle has a chord that goes from #( 5 pi)/6 # to #(7 pi) / 4 # radians on the circle. If the area of the circle is #20 pi #, what is the length of the chord?

Question

A circle has a chord that goes from #( 5 pi)/6  # to #(7  pi) /  4  # radians on the circle.  If the area of the circle is #20   pi  #, what is the length of the chord?

Caroline Aucoin · Accepted Answer

#8.87" to 2 dec. places"# #"we require to use the "color(blue)"cosine rule"# #color(red)(bar(ul(|color(white)(2/2)color(black)(a^2=b^2+c^2-(2bc cosA)color(white)(2/2)|)))# #"we also require the radius of the circe"# #• " area "=pir^2=20pi# #rArrr^2=20rArrr=sqrt20=2sqrt5# #"the angle subtended at the centre of the circle by the chord is"# #angleA=(7pi)/4-(5pi)/6=(21pi)/12-(10pi)/12=(11pi)/12# #"we now have a triangle formed by the 2 radii and the chord"# #"using the "color(blue)"cosine rule"# #"with "a=" chord ",b" and "c=" radii"# #"and A the angle between the radii"# #rArra^2=(2sqrt5)^2+(2sqrt5)^2-(2xx2sqrt5xx2sqrt5xxcos((11pi)/12))# #color(white)(rArra^2)=20+20-(40xxcos((11pi)/12))# #color(white)(rArra^2)=78.6370..# #rArr"length of chord "=sqrt(78.637...)~~8.87#

Parker Allen · Answer

undefined To find the length of the chord, we need to use the formula for the length of a chord in a circle, which is given by $2r \sin(\frac{\theta}{2})$, where $r$ is the radius of the circle and $\theta$ is the angle subtended by the chord at the center of the circle.

First, we need to find the radius of the circle. We know that the area of the circle is $20\pi$, so we can use the formula for the area of a circle, $A = \pi r^2$, to find the radius:
$$20\pi = \pi r^2$$
$$r^2 = 20$$
$$r = \sqrt{20} = 2\sqrt{5}$$

Next, we need to find the angle $\theta$ subtended by the chord at the center of the circle. The angle $\theta$ can be found by taking the difference between the given angles:
$$\theta = (\frac{7\pi}{4}) - (\frac{5\pi}{6})$$
$$\theta = \frac{14\pi}{12} - \frac{10\pi}{12}$$
$$\theta = \frac{4\pi}{12} = \frac{\pi}{3}$$

Now, we can use the formula for the length of a chord to find the length of the chord:
$$Length = 2r \sin(\frac{\theta}{2})$$
$$Length = 2(2\sqrt{5}) \sin(\frac{\pi}{6})$$
$$Length = 4\sqrt{5} \cdot \frac{1}{2}$$
$$Length = 2\sqrt{5}$$

So, the length of the chord is $2\sqrt{5}$ units.