# A triangle has corners at (5 ,4 ), (9 ,3 ), and (2 ,7 ). How far is the triangle's centroid from the origin?

Oct 27, 2016

The distance is $\frac{\sqrt{452}}{3}$

#### Explanation:

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

So the centroid is $\left(\frac{5 + 9 + 2}{3} , \frac{4 + 3 + 7}{3}\right) = \left(\frac{16}{3} , \frac{14}{3}\right)$
Let the centroid be $\left({x}_{c} , {y}_{c}\right)$
The distance of the centroid from the origin is $\sqrt{{x}_{c}^{2} + {y}_{c}^{2}}$

$= \frac{\sqrt{{16}^{2} + {14}^{2}}}{3} = \frac{\sqrt{452}}{3}$