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?

Question

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?

Evelyn Arboleda · Accepted Answer

≈ 8.378 units.

The length of the arc is found by calculating the fraction of the circumference. That is #color(red)(bar(ul(|color(white)(2/2)color(black)("arc" ="circumference" xxx/(2pi))color(white)(2/2)|)))# where #x# is the angle subtended at the centre of the circle. To calculate circumference #=2pir# we require to know r, the radius. 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. #rArr"radius" =r=5-1=4# length of arc #=cancel(2pi)xx4xx((2pi)/3)/(cancel(2pi))# #=(4xx2pi)/3≈8.378" units to 3 decimal places"#

Evelyn Arboleda · Answer

undefined 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.