# A triangle has corners at (8, 1 ), ( 5, -8), and (7, -2 )#. If the triangle is reflected across the x-axis, what will its new centroid be?

Dec 6, 2016

The coordinates of the centroid of the triangle after it has been reflected over the $x$ axis are $\left(\frac{20}{3} , 3\right)$.

#### Explanation:

The reflection of $\left(x , y\right)$ over the $x$ axis is $\left(x , - y\right)$.

If a triangle has vertices of $\left(8 , 1\right) , \left(5 , - 8\right) , \left(7 , - 2\right)$,

the vertices after reflecting it over the $x$ axis are $\left(8 , - 1\right) , \left(5 , 8\right) , \left(7 , 2\right)$.

To find the coordinates of a centroid of a triangle with vertices at $\left({x}_{1} , {y}_{1}\right) , \left({x}_{2} , {y}_{2}\right) , \left({x}_{3} , {y}_{3}\right)$, use the formula

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

${\left(x , y\right)}_{\text{centroid}} = \left(\frac{8 + 5 + 7}{3} , \frac{- 1 + 8 + 2}{3}\right) = \left(\frac{20}{3} , 3\right)$