# A triangle has corners at (1 ,9 ), (7 ,8 ), and (4 ,5 ). How far is the triangle's centroid from the origin?

Jun 5, 2018

$\textcolor{b l u e}{\frac{2 \sqrt{697}}{3} \text{ units}}$

#### Explanation:

The centroid can be found by taking the arithmetic mean of the $\boldsymbol{x}$ and $\boldsymbol{y}$ coordinates of the triangles vertices:

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

So we have:

$\left(\frac{1 + 7 + 4}{3} , \frac{9 + 8 + 5}{3}\right) \implies \left(4 , \frac{22}{3}\right)$

Distance from the origin can be found using the distance formula:

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

$| d | = \sqrt{{\left(4 - 0\right)}^{2} + {\left(\frac{22}{3} - 0\right)}^{2}} = \frac{2 \sqrt{697}}{3}$ units