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

Claire Akin · Accepted Answer

≈ 19.83

To calculate the length of arc , require to know radius of circle. This can be found using the #color(blue) " distance formula "# # d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2# where#(x_1,y_1)" and " (x_2,y_2) " are 2 coord points "# The 2 points here are the centre and the point it passes through. This distance is the radius of the circle. let#(x_1,y_1)=(7,5)" and " (x_2,y_2)=(5,8) # hence r #=sqrt((5-7)^2+(8-5)^2)=sqrt(4+9)=sqrt13# arc length = circumference #xx " fraction of circle covered "# arc length =#2pirxx((7pi)/4)/(2pi) = cancel((2pi)r)xx((7pi)/4)/cancel(2pi) =rxx(7pi)/4# #rArr" arc length " = sqrt13xx(7pi)/4 ≈ 19.83 #

Caroline Aucoin · Answer

undefined First, find the radius of the circle using the distance formula. Then, use the formula for the length of an arc of a circle, which is given by $ \text{arc length} = r \cdot \theta $, where $ r $ is the radius of the circle and $ \theta $ is the central angle in radians.

Given the coordinates of the center and a point on the circle, you can find the radius using the distance formula. Then, you can calculate the arc length using the formula mentioned above.

Let's proceed with the calculations:

1. Calculate the radius using the distance formula:  
   $ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $

$ r = \sqrt{(5 - 7)^2 + (8 - 5)^2} $  
   $ r = \sqrt{(-2)^2 + (3)^2} $  
   $ r = \sqrt{4 + 9} $  
   $ r = \sqrt{13} $

2. The arc length is given by:  
   $ \text{arc length} = r \cdot \theta $

$ \text{arc length} = \sqrt{13} \times \frac{7\pi}{4} $

$ \text{arc length} = \frac{7\pi\sqrt{13}}{4} $