A circle's center is at #(7 ,5 )# and it passes through #(5 ,4 )#. What is the length of an arc covering #(5pi ) /3 # radians on the circle?

Question

A circle's center is at #(7   ,5   )# and it passes through #(5    ,4   )#.  What is the length of an arc covering #(5pi ) /3  # radians on the circle?

Mason Adams · Accepted Answer

# ( 5 sqrt(5) pi ) / 3 # Please check my solution here for a detailed explanation. Solution Radius # r = sqrt( (5 - 7)^2 + (4 - 5)^2 ) = sqrt(5) #. Arc length # S = ( 5 sqrt(5) pi ) / 3 #.

Sadie Adams · Answer

undefined To find the length of an arc covering $ \frac{5\pi}{3} $ radians on the circle, we first need to find the radius of the circle using the given center and a point on the circle.

Given: Center of the circle $ (7, 5) $ and a point on the circle $ (5, 4) $

We can use the distance formula to find the distance between the center and the point on the circle, which will give us the radius.

$$ \text{Radius} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

$$ \text{Radius} = \sqrt{(5 - 7)^2 + (4 - 5)^2} $$

$$ \text{Radius} = \sqrt{(-2)^2 + (-1)^2} $$

$$ \text{Radius} = \sqrt{4 + 1} $$

$$ \text{Radius} = \sqrt{5} $$

Now, we can use the formula for the length of an arc of a circle:

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

$$ \text{Length of arc} = \sqrt{5} \times \frac{5\pi}{3} $$

$$ \text{Length of arc} = \frac{5\sqrt{5}\pi}{3} $$

Thus, the length of the arc covering $ \frac{5\pi}{3} $ radians on the circle is $ \frac{5\sqrt{5}\pi}{3} $.