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

Mar 11, 2016

≈ 1.756 units

Explanation:

To calculate length of arc , the radius is required. This can be found using the 2 points given and the $\textcolor{b l u e}{\text{ distance formula }}$

$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

where $\left({x}_{1} , {y}_{1}\right) \text{ and "(x_2,y_2)" are 2 coordinate points }$

let $\left({x}_{1} , {y}_{1}\right) = \left(1 , 3\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(2 , 1\right)$

radius (r ) $= \sqrt{{\left(2 - 1\right)}^{2} + {\left(1 - 3\right)}^{2}} = \sqrt{5}$

length of arc = $2 \pi r \times \text{fraction of circle covered }$

 = cancel(2pi)xxsqrt5 xx (pi/4)/cancel(2pi) = sqrt5xxpi/4 ≈ 1.756