A circle's center is at #(5 ,4 )# and it passes through #(1 ,4 )#. What is the length of an arc covering #(2pi ) /3 # radians on the circle?

The length of the arc is found by calculating the fraction of the circumference.

We are given the coordinates of the centre and a point on the circumference. Hence the radius is the distance between these 2 points. The 2 points (5 ,4) and (1 ,4) have the same y-coordinate and so they lie on a horizontal line ( y = 4). The distance between the points is therefore the difference in the x-coordinates.

To find the length of an arc covering $\frac{2\pi}{3}$ radians on a circle, you need the circle's radius and the formula for arc length: $L = r\theta$, where $L$ is the arc length, $r$ is the radius, and $\theta$ is the angle in radians.

1. Find the Radius: The distance between the center of the circle $(5, 4)$ and a point it passes through $(1, 4)$ will give you the radius. Since the y-coordinates are the same, the distance is just the difference in the x-coordinates, which is $|5 - 1| = 4$. So, the radius $r = 4$.

2. Calculate the Arc Length: Using the formula $L = r\theta$ with $r = 4$ and $\theta = \frac{2\pi}{3}$,

$L = 4 \times \frac{2\pi}{3}$

Let's calculate the exact length of the arc.The length of an arc covering $\frac{2\pi}{3}$ radians on the circle is approximately 8.38 units.