# A triangle has corners at (3 ,5 ), (4 ,7 ), and (1 ,2 ). How far is the triangle's centroid from the origin?

May 17, 2016

The distance of triangle's centroid from the origin is $5.375$ units

#### Explanation:

The centroid of a triangle whose vertices are $\left({x}_{1} , {y}_{1}\right)$, $\left({x}_{2} , {y}_{2}\right)$ and (x_3,y_3)# is given by

$\left(\frac{{x}_{1} + {x}_{2} + {x}_{3}}{3} , \frac{{y}_{1} + {y}_{2} + {y}_{3}}{3}\right)$

Hence centrid of given triangle is $\left(\frac{3 + 4 + 1}{3} , \frac{5 + 7 + 2}{3}\right)$ or $\left(\frac{8}{3} , \frac{14}{3}\right)$

and its distance from origin is

$\sqrt{{\left(\frac{8}{3}\right)}^{2} + {\left(\frac{14}{3}\right)}^{2}} = \sqrt{\frac{64}{9} + \frac{196}{9}} = \sqrt{\frac{260}{9}} = \frac{1}{3} \times 16.125 = 5.375$