A circle's center is at #(2 ,4 )# and it passes through #(1 ,2 )#. What is the length of an arc covering #(5pi ) /4 # radians on the circle?

Question

A circle's center is at #(2 ,4   )# and it passes through #(1    ,2   )#.  What is the length of an arc covering #(5pi ) /4  # radians on the circle?

Theodore Abad · Accepted Answer

Length of the Arc #=8.8# Circle's center is at #(2,4)# and it passes through #(1,2)# Therefore Length of the #radius=r# =Distance between these points#(2,4) and (1,2)# or #radius =r=sqrt((2-1)^2+(4-2)^2)# #=sqrt(1^2+2^2)# #=sqrt(1+4)# #=sqrt5# #=2.24# Therefore Circumference of the Circle #=2pir=2pitimes2.4=14.07# Arc covers #(5pi)/4# radians on the Circle In other words Arc covers #(5pi)/4-:2pi=5/8times #(circumference of the Circle) Therefore Length of the Arc #=5/8 times2pir=5/8 times14.07=8.8#

Allison Allen · Answer

undefined To find the length of an arc covering $ \frac{5\pi}{4} $ radians on the circle, we need to first find the radius of the circle. Using the distance formula, we find the distance between the center of the circle (2, 4) and the point on the circle (1, 2). Then, we use the formula for the length of an arc on a circle:

$$ \text{Arc Length} = r \times \text{angle in radians} $$

where $ r $ is the radius of the circle.

The distance between the center of the circle (2, 4) and the point on the circle (1, 2) is:

$$ \sqrt{(1 - 2)^2 + (2 - 4)^2} = \sqrt{1 + 4} = \sqrt{5} $$

So, the radius of the circle is $ \sqrt{5} $.

Now, we can find the length of the arc:

$$ \text{Arc Length} = \sqrt{5} \times \frac{5\pi}{4} = \frac{5\sqrt{5}\pi}{4} $$

Therefore, the length of the arc covering $ \frac{5\pi}{4} $ radians on the circle is $ \frac{5\sqrt{5}\pi}{4} $.