# A triangle has corners at (1 ,5 ), (9 ,4 ), and (6 ,7 ). How far is the triangle's centroid from the origin?

Feb 5, 2018

$O G = \textcolor{b l u e}{4 \sqrt{\frac{2}{3}} \approx 3.266}$

#### Explanation:

The centroid of a triangle is the point of intersection of its medians. Centroid divides the medians in the ratio 2 : 1.

If the vertices of a triangle are (x1,y1), (x2,y2), (x3-y3), then the centroid of the triangle is

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

${G}_{x} = \frac{1 + 9 + 6}{3} = \frac{16}{3}$

${G}_{y} = \frac{5 + 4 + 7}{3} = \frac{16}{3}$

Distance of G from origin O is given by the distance formula,

$O G = \sqrt{{\left({G}_{x} - {O}_{x}\right)}^{2} + {\left({G}_{y} - {O}_{y}\right)}^{2}} = \sqrt{{\left(\left(\frac{16}{3}\right) - 0\right)}^{2} + \left({\left(\frac{16}{3} - 0\right)}^{2}\right)}$

$O G = \sqrt{\frac{32}{3}} = \textcolor{b l u e}{4 \sqrt{\frac{2}{3}} \approx 3.266}$