A triangle has corners at (1 ,9 ), (5 ,4 ), and (3 ,8 ). How far is the triangle's centroid from the origin?

Jun 18, 2016

It is $7.61$.

Explanation:

If $\left({A}_{x} , {A}_{y}\right) , \left({B}_{x} , {B}_{y}\right) , \left({C}_{x} , {C}_{y}\right)$ are the vertex of a triangle, the coordinates of the centroid are

$C {O}_{x} = \frac{{A}_{x} + {B}_{x} + C + x}{3}$

$C {O}_{y} = \frac{{A}_{y} + {B}_{y} + C + y}{3}$

In our case

$C {O}_{x} = \frac{1 + 5 + 3}{3} = \frac{9}{3} = 3$

$C {O}_{y} = \frac{9 + 4 + 8}{3} = \frac{21}{3} = 7$

Then the centroid has coordinates $\left(3 , 7\right)$ and its distance from the origin is simply

$d = \sqrt{{3}^{2} + {7}^{2}} = \sqrt{9 + 49} = \sqrt{58} \setminus \approx 7.61$.