# A triangle has corners at (5 ,1 ), (7 ,2 ), and (6 ,7 ). How far is the triangle's centroid from the origin?

Nov 13, 2016

The distance between the centroid and the origin is $d = \frac{2}{3} \sqrt{106}$

#### Explanation:

The centroid of a triangle can be found by averaging the $x$ coordinates and $y$ coordinates of the vertices.

$x = \frac{5 + 7 + 6}{3} = \frac{18}{3} = 6$

$y = \frac{1 + 2 + 7}{3} = \frac{10}{3}$

The coordinates of the centroid are $\left(6 , \frac{10}{3}\right)$

To find the distance from the origin, use the distance formula.

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

$d = \sqrt{{\left(6 - 0\right)}^{2} + {\left(\frac{10}{3} - 0\right)}^{2}}$

$d = \sqrt{36 + \frac{100}{9}}$

$d = \sqrt{\frac{424}{9}}$

$d = \sqrt{\frac{4 \cdot 106}{9}}$

$d = \frac{2}{3} \sqrt{106}$