# A triangle has corners at (8, 3 ), ( 5, -8), and (7, -4 )#. If the triangle is reflected across the x-axis, what will its new centroid be?

Oct 23, 2016

$\left(\frac{20}{3} , 3\right)$

#### Explanation:

The first step is to find the coordinates of the centroid, $\left({x}_{c} , {y}_{c}\right)$

Given that the vertices of a triangle are $\left({x}_{1} , {y}_{1}\right) , \left({x}_{2} , {y}_{2}\right) , \left({x}_{3} , {y}_{3}\right)$

Then.

${x}_{c} = \frac{1}{3} \left({x}_{1} + {x}_{2} + {x}_{3}\right) \text{ the average of the x-coordinates}$

and ${y}_{c} = \frac{1}{3} \left({y}_{1} + {y}_{2} + {y}_{3}\right) \text{ the average of the y-coordinates}$

Here.

$\left({x}_{1} , {y}_{1}\right) = \left(8 , 3\right) , \left({x}_{2} , {y}_{2}\right) = \left(5 , - 8\right) , \left({x}_{3} , {y}_{3}\right) = \left(7 , - 4\right)$

$\Rightarrow {x}_{c} = \frac{1}{3} \left(8 + 5 + 7\right) = \frac{20}{3}$

and ${y}_{c} = \frac{1}{3} \left(3 - 8 - 4\right) = - 3$

coordinates of centroid $= \left(\frac{20}{3} , - 3\right)$

Under reflection in the x-axis, a point (x ,y) → (x ,-y)

$\Rightarrow \left(\frac{20}{3} , - 3\right) \to \left(\frac{20}{3} , 3\right)$