# A triangle has corners at (3 ,4 ), (4 ,7 ), and (2 ,8 ). How far is the triangle's centroid from the origin?

Apr 27, 2016

≈ 7 units

#### Explanation:

The first step here is to find the coordinates of the centroid
( x_c" and " y_c)

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

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

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

here

${x}_{c} = \frac{1}{3} \left(3 + 4 + 2\right) = 3 \text{ and } {y}_{c} = \frac{1}{3} \left(4 + 7 + 8\right) = \frac{19}{3}$

coordinates of centroid $= \left(3 , \frac{19}{3}\right)$

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

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

The 2 points here are the centroid and the origin
Since the origin (0,0) is one of the points this simplifies the distance formula to

 d =sqrt(3^2 + (19/3)^2)=sqrt(9+361/9) ≈ 7 "units"