# A triangle has corners at (1 ,9 ), (3 ,4 ), and (4 ,5 ). How far is the triangle's centroid from the origin?

Sep 30, 2016

≈ 6.566 units.

#### Explanation:

Given $\left({x}_{1} , {y}_{1}\right) , \left({x}_{2} , {y}_{2}\right) \text{ and } \left({x}_{3} , {y}_{3}\right)$ are the vertices of a triangle then the coordinates of the centroid are.

${x}_{c} = \frac{1}{3} \left({x}_{1} + {x}_{2} + {x}_{3}\right) \text{ and } {y}_{c} = \frac{1}{3} \left({y}_{1} + {y}_{2} + {y}_{3}\right)$

That is the average of the coordinates of the vertices.

substituting the given coordinates into the above.

${x}_{c} = \frac{1}{3} \left(1 + 3 + 4\right) = \frac{8}{3} \text{ and } {y}_{c} = \frac{1}{3} \left(9 + 4 + 5\right) = 6$

$\Rightarrow \text{coordinates of centroid} = \left(\frac{8}{3} , 6\right)$

To calculate the distance from this point to the origin use $\textcolor{b l u e}{\text{Pythagoras' theorem}}$

$d = \sqrt{{\left(\frac{8}{3}\right)}^{2} + {6}^{2}} = \sqrt{\frac{64}{9} + \frac{324}{9}}$

=sqrt(388/9)≈6.566" to 3 decimal places"