Question

A triangle has corners at `(3, 9 )`, ( 6, -5)`, and`( 4, -1)#. If the triangle is reflected across the x-axis, what will its new centroid be?

Answer 1

`(13/3 , -1 )`

Explanation:

The first step is to find the coordinates of the centroid.

Given the 3 vertices of a triangle `(x_1,y_1) , (x_2,y_2) , (x_3,y_3)`

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

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

Here let `(x_1,y_1)=(3,9),(x_2,y_2)=(6,-5),(x_3,y_3)=(4,-1)`

Hence coords of centroid

`= [1/3(3+6+4) , 1/3(9-5-1) ] = (13/3 , 1)`

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

new centroid `(13/3 , 1) → (13/3 , -1)`