Question

A circle has a center at `(7 ,6 )` and passes through `(2 ,1 )`. What is the length of an arc covering `pi/8` radians on the circle?

Answer 1

`( 5 sqrt(2) pi ) / 8`

Explanation:

We know the center and a point on the circle. The distance between the two points is the radius (draw a picture and convince yourself this).

We know that the distance between two points `(x_1, y_1)` and `(x_2, y_2)` on the Euclidean plane is given by `d = sqrt( (x_1 - x_2)^2 + (y_1 - y_2)^2 )`.
Thus, radius `r = sqrt( (7 - 2)^2 + (6 - 1)^2 ) = sqrt(5^2 + 5^2) = sqrt(2 * 5^2) = 5 sqrt(2)`.

By definition, a radian is the angle subtended by an arc of length equal to the radius. Thus, an arc which subtends an angle of `theta` has a length of `theta r`. In this case, `theta = pi / 8`, so arc length = `( 5 sqrt(2) pi ) / 8`.

Hint to remember
Note that the circumference subtends and angle of `2 pi` at the center, and has a length of `2 pi r`. By unitary method, an arc which subtends an angle of `theta` has a length of `theta r`.