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?

Answer 1

≈ 8.378 units.

Explanation:

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"`