A circle's center is at #(7 ,2 )# and it passes through #(5 ,8 )#. What is the length of an arc covering #(7pi ) /4 # radians on the circle?

Question

A circle's center is at #(7   ,2   )# and it passes through #(5    ,8   )#.  What is the length of an arc covering #(7pi ) /4  # radians on the circle?

Claire Akin · Accepted Answer

#=34.76# So the distance between points #(7,2)# and #(5,8)# is the radius of the circle. So the distance is #=sqrt((7-5)^2+(8-2)^2# #=sqrt(2^2+6^2)# #=sqrt(4+36)# #=sqrt40# #=6.32# Therefore radius of the Circle #r=6.32# The Circumference of the circle#=2pir=2pi(6.32)=39.73# An Arc covers #7pi/4# radians or It covers #7pi/4-:2pi=7/8# of the Circumference Therefore; Length of the Arc#=7/8(39.73)=34.76#

Jayden Jackson · Answer

undefined To find the length of an arc covering $ \frac{7\pi}{4} $ radians on the circle, first, calculate the radius of the circle using the given center and a point it passes through, then use the formula for arc length.

1. Calculate the distance between the center of the circle $(7, 2)$ and the point it passes through $ (5, 8) $ using the distance formula.
$$ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

2. Use the distance found as the radius of the circle.

3. Use the formula for arc length:
$$ \text{Arc Length} = \text{radius} \times \text{angle in radians} $$

4. Substitute the values to find the arc length.