# A triangle has corners at (2 ,4 ), (8 ,6 ), and (7 ,1 ). How far is the triangle's centroid from the origin?

Nov 11, 2016

The distance is$= \frac{\sqrt{410}}{3}$

#### Explanation:

Let the corners of a triangle be $\left({x}_{1} , {y}_{1}\right) , \left({x}_{2} , {y}_{2}\right) \mathmr{and} \left({x}_{3} , {y}_{3}\right)$

And the centrid $\left({x}_{c} , {y}_{c}\right)$

Then ${x}_{c} = \frac{{x}_{1} + {x}_{2} + {x}_{3}}{3} = \frac{2 + 8 + 7}{3} = \frac{17}{3}$

${y}_{c} = \frac{{y}_{1} + {y}_{2} + {y}_{3}}{3} = \frac{4 + 6 + 1}{3} = \frac{11}{3}$

The distance of the centroid from the origin is
$d = \sqrt{{x}_{c}^{2} + {y}_{c}^{2}} = \sqrt{{\left(\frac{17}{3}\right)}^{2} + {\left(\frac{11}{3}\right)}^{2}} = \frac{\sqrt{410}}{3}$