# A triangle has corners at (6, 4 ), ( 2, 5), and ( 7, -1)#. If the triangle is reflected across the x-axis, what will its new centroid be?

Jun 13, 2017

Coordinates of new centroid are $\left(5 , - \frac{8}{3}\right)$.

#### Explanation:

The coordinates of the centroid of a triangle, whose corners are $\left({x}_{1} , {y}_{1}\right)$, $\left({x}_{2} , {y}_{2}\right)$ and $\left({x}_{3} , {y}_{3}\right)$, are $\left(\frac{{x}_{1} + {x}_{2} + {x}_{3}}{3} , \frac{{y}_{1} + {y}_{2} + {y}_{3}}{3}\right)$.

As such coordinates of centroid of the triangle with corners at $\left(6 , 4\right)$, $\left(2 , 5\right)$ and $\left(7 , - 1\right)$ are $\left(\frac{6 + 2 + 7}{3} , \frac{4 + 5 - 1}{3}\right)$ i.e. $\left(5 , \frac{8}{3}\right)$.

When a point $\left(x , y\right)$ is reflected across $x$-axis, its coordinates change to $\left(x , - y\right)$.

Hence new coordinates of corners of the triangle are $\left(6 , - 4\right)$, $\left(2 , - 5\right)$ and $\left(7 , 1\right)$ and coordinates of new centroid are $\left(\frac{6 + 2 + 7}{3} , \frac{- 4 - 5 + 1}{3}\right)$ i.e. $\left(5 , - \frac{8}{3}\right)$.

Observe that centroid too is reflected across $x$-axis.