A circle's center is at #(2 ,6 )# and it passes through #(3 ,1 )#. What is the length of an arc covering #(pi ) /3 # radians on the circle?

Question

A circle's center is at #(2 ,6 )# and it passes through #(3    ,1     )#.  What is the length of an arc covering #(pi ) /3  # radians on the circle?

Evelyn Arboleda · Accepted Answer

arc length = radius x arc angle
radius = #sqrt(26)#
arc angle = #(pi)/3#

arc length = #sqrt(26)((pi)/3) ~= 5.34# First find the radius, r, using the equation of a circle: #(x-x_0)^2+(y-y_0)^2=r^2# The circle's centre is (2,6), so #x_0=2 and y_0=6# The circle passes through (3,1), so at this point x=3 and y=1. Inputting all this into the circle equation: #(3-2)^2+(1-6)^2=r^2# #(1)^2+(-5)^2=r^2# #1+25=r^2=26# Therefore: r = #sqrt(26)# Now we can find the arc length using: arc length = radius x arc angle arc angle = #(pi)/3# arc length = #sqrt(26)((pi)/3) ~= 5.34#

Claire Akin · Answer

undefined To find the length of an arc covering $ \frac{\pi}{3} $ radians on the circle, follow these steps:

1. Determine the radius of the circle using the given center and one point it passes through, using the distance formula.
2. Use the formula for arc length $ s = r \cdot \theta $, where $ r $ is the radius of the circle and $ \theta $ is the angle in radians subtended by the arc.
3. Substitute the values of $ r $ and $ \theta $ into the formula and compute the length of the arc.

By following these steps, you can find the length of the arc covering $ \frac{\pi}{3} $ radians on the circle.