Question

A triangle has corners at `(9, 4 )`, ( 5, -9)`, and`( 2, -3)#. If the triangle is reflected across the x-axis, what will its new centroid be?

Answer 1

`(16/3,8/3)`

Explanation:

The first step is to calculate the coordinates of the existing centroid.
Given the 3 vertices of a triangle

`(x_1,y_1),(x_2,y_2)" and " (x_3,y_3)`

x-coordinate of centroid =`color(red)(|bar(ul(color(white)(a/a)color(black)(1/3(x_1+x_2+x_3))color(white)(a/a)|)))`

y-coordinate of centroid = `color(red)(|bar(ul(color(white)(a/a)color(black)(1/3(y_1+y_2+y_3))color(white)(a/a)|)))`

Basically, this is the average of the x and y-coordinates of the vertices.

Thus x-coordinate = `1/3(9+5+2)=16/3`

and y-coordinate = `1/3(4-9-3)=-8/3`

coordinates of centroid `=(16/3,-8/3)`

Under a reflection in the x-axis

a point (x ,y) → (x ,-y)

hence `(16/3,-8/3)to(16/3,8/3)" new centroid"`