# A triangle has corners at (9 ,3 ), (6 ,7 ), and (3 ,2 ). How far is the triangle's centroid from the origin?

May 10, 2017

Centroid is $\text{at } \left(6 , 4\right)$; distance to $\left(0 , 0\right)$ is $2 \sqrt{13} \approx 7.2$

#### Explanation:

To find the centroid you need to find the midpoints using $\left(\frac{{x}_{1} + {x}_{2}}{2} , \frac{{y}_{1} + {y}_{2}}{2}\right)$

Midpoint between $\left(3 , 2\right) \text{ and } \left(9 , 3\right) = \left(6 , 2.5\right)$

Midpoint between $\left(3 , 2\right) \text{ and } \left(6 , 7\right) = \left(4.5 , 4.5\right)$

Midpoint between $\left(6 , 7\right) \text{ and } \left(9 , 3\right) = \left(7.5 , 5\right)$

Connect the midpoints to the angle opposite. The intersection is the centroid:

The centroid is found at $\left(6 , 4\right)$

The distance from $\left(6 , 4\right) \text{ and } \left(0 , 0\right) = \sqrt{{6}^{2} + {4}^{2}} = \sqrt{52} = 2 \sqrt{13} \approx 7.2$