A circle's center is at #(5 ,9 )# and it passes through #(7 ,3 )#. What is the length of an arc covering #(15pi ) /8 # radians on the circle?

Question

A circle's center is at #(5 ,9 )# and it passes through #(7    ,3    )#.  What is the length of an arc covering #(15pi ) /8  # radians on the circle?

Autumn Armstrong · Accepted Answer

#(15sqrt10pi)/4~~37.25# units

Sophia Alfred · Answer

undefined To find the length of an arc covering $ \frac{15\pi}{8} $ radians on the circle, you can use the formula:

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

Where:
- $ r $ is the radius of the circle.
- Angle in radians is $ \frac{15\pi}{8} $.

To find the radius $ r $, you can use the distance formula between the center of the circle and a point on the circle:

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

Where:
- $ (x_1, y_1) $ is the center of the circle, given as (5, 9).
- $ (x_2, y_2) $ is the point on the circle, given as (7, 3).

$$ r = \sqrt{(7 - 5)^2 + (3 - 9)^2} = \sqrt{2^2 + (-6)^2} = \sqrt{40} $$

$$ r = 2\sqrt{10} $$

Now, you can plug in the values into the arc length formula:

$$ \text{Arc Length} = 2\sqrt{10} \times \frac{15\pi}{8} $$

$$ \text{Arc Length} = \frac{15\pi}{4}\sqrt{10} $$

So, the length of the arc covering $ \frac{15\pi}{8} $ radians on the circle is $ \frac{15\pi}{4}\sqrt{10} $.