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

May 18, 2018

$\left(\frac{11}{3} , \frac{4}{3}\right)$

#### Explanation:

$\text{begin by calculating the coordinates of the centroid}$

$\text{given the coordinates of the vertices of a triangle say}$

$\left({x}_{1} , {y}_{1}\right) , \left({x}_{2} , {y}_{2}\right) \text{ and } \left({x}_{3} , {y}_{3}\right)$

$\text{then the centroid is the average of the x and y}$
$\text{coordinates of the vertices}$

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

$\Rightarrow \left[\frac{1}{3} \left(8 + 2 + 1\right) , \frac{1}{3} \left(5 - 7 - 2\right)\right] = \left(\frac{11}{3} , - \frac{4}{3}\right)$

$\text{under a reflection in the x-axis}$

• " a point "(x,y)to(x,-y)#

$\Rightarrow \left(\frac{11}{3} , - \frac{4}{3}\right) \to \left(\frac{11}{3} , \frac{4}{3}\right) \leftarrow \textcolor{red}{\text{new centroid}}$