# A triangle has corners at (7 ,6 ), (4 ,3 ), and (5 ,2 ). How far is the triangle's centroid from the origin?

Mar 21, 2017

See below

#### Explanation:

To find the centroid of a triangle, we just find the average of the $x$-vertices, and then the average of the $y$-vertices, or ${x}_{c e n t r o i d} = \frac{{x}_{1} + {x}_{2} + {x}_{3}}{3}$
${y}_{c e n t r o i d} = \frac{{y}_{1} + {y}_{2} + {y}_{3}}{3}$

The centroid is just:

${x}_{c e n t r o i d} = \frac{7 + 4 + 5}{3} = \frac{16}{3}$ and
${y}_{c e n t r o i d} = \frac{6 + 3 + 2}{3} = \frac{11}{3}$

Then, the distance from the origin to this point is:

distance $= \sqrt{{\left(\frac{16}{3}\right)}^{2} + {\left(\frac{11}{3}\right)}^{2}} = \sqrt{\frac{377}{9}} = \frac{\sqrt{377}}{3}$