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

May 9, 2016

≈ 5.897

#### Explanation:

The first step is to find the coordinates of the centroid.

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

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

y-coord$\left({y}_{c}\right) = \frac{1}{3} \left({y}_{1} + {y}_{2} + {y}_{3}\right)$
so ${x}_{c} = \frac{1}{3} \left(5 + 4 + 3\right) = 4 \text{ and } {y}_{c} = \frac{1}{3} \left(2 + 6 + 5\right) = \frac{13}{3}$

coords of centroid $= \left(4 , \frac{13}{3}\right)$

To calculate the distance the centroid is from the origin use the $\textcolor{b l u e}{\text{ distance formula }}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
where $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 points }$

The 2 points here are (0,0) and $\left(4 , \frac{13}{3}\right)$

$d = \sqrt{{\left(4 - 0\right)}^{2} + {\left(\frac{13}{3} - 0\right)}^{2}} = \sqrt{16 + \frac{169}{9}}$

=sqrt(144/9+169/9)=sqrt(313/9) ≈ 5.897