# A triangle has corners at (3, 9 ), ( 7, -2), and ( 5, -9 )#. If the triangle is reflected across the x-axis, what will its new centroid be?

Jul 28, 2017

The new coordinates of the centroid is $= \left(5 , \frac{2}{3}\right)$

#### Explanation:

Let the corners of the triangle be $\left({x}_{1} , {y}_{1}\right)$, $\left({x}_{2} , {y}_{2}\right)$ and $\left({x}_{3} , {y}_{3}\right)$

The coordinates of the centroid are

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

Here, we have $\left(3 , 9\right)$, $\left(7 , - 2\right)$, and $\left(5 , - 9\right)$

So,

The coordinates of the centroid are $C = \left(\frac{3 + 7 + 5}{3} , \frac{9 - 2 - 9}{3}\right) = \left(\frac{15}{3} , - \frac{2}{3}\right) = \left(5 , - \frac{2}{3}\right)$

The matrix of the reflection acrossthe x-axis is

$M = \left(\begin{matrix}1 & 0 \\ 0 & - 1\end{matrix}\right)$

Therefore,

The new coordinates of the centroid is

$\left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}1 & 0 \\ 0 & - 1\end{matrix}\right) \left(\begin{matrix}5 \\ - \frac{2}{3}\end{matrix}\right) = \left(\begin{matrix}5 \\ \frac{2}{3}\end{matrix}\right)$