# A triangle has corners at (9 ,3 ), (7 ,5 ), and (3 ,1 ). How far is the triangle's centroid from the origin?

Dec 13, 2016

The answer is $= 7$

#### Explanation:

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

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

Therefore,

we have ${x}_{g} = \frac{9 + 7 + 3}{3} = \frac{19}{3}$

and ${y}_{g} = \frac{3 + 5 + 1}{3} = 3$

The distance of $\left({x}_{g} , {y}_{g}\right)$ from the origin is

$= \sqrt{{x}_{g}^{2} + {y}_{g}^{2}} = \sqrt{{19}^{2} / 9 + 9} = 7$