Question

A circle has a center at `(3 ,5 )` and passes through `(0 ,2 )`. What is the length of an arc covering `(3pi ) /4` radians on the circle?

Answer 1

`l_a=9/4sqrt2pi`.

Explanation:

The radius of the circle with center in `C(3,5)` that passes through `A(0,2)` is the distance between the two points:

`r=sqrt((x_A-x_C)^2+(y_A-y_C)^2)=`

`=sqrt((0-3)^2+(2-5)^2)=sqrt(9+9)=sqrt(2*9)=3sqrt2`.

The angle at the center of a circle is, obviously, `2pi`, so the angle of `3/4pi` is: `(3/4pi)/(2pi)=3/8` of the whole circle.

Since the lenght of a circle is: `l_c=2pir=2pi*3sqrt2=6sqrt2pi`, then the lenght of the arc is:

`l_a=6sqrt2pi*3/8=9/4sqrt2pi`.