A triangle has corners at #(4, 1 )#, ( 6, -8)#, and #(3, -2 )#. If the triangle is reflected across the x-axis, what will its new centroid be?

Question

A triangle has corners at #(4, 1  )#, ( 6,  -8)#, and #(3,  -2  )#.  If the triangle is reflected across the x-axis, what will its new centroid be?

Sadie Adams · Accepted Answer

New centroid is #(13/3,3)#

The centroid of the triangle with corners at #(4,1)#, #(6,-8)# and #(3, -2 )# is

#((4+6+3)/3,(1+(-8)+(-2))/3)# or #(13/3,-3)#

When triangle is reflected across the x-axis, its centroid too is reflected across the x-axis

and as reflection of a point #(x_1,y_1)# in x-axis is #(x_1,-y_1)#,

New centroid is #(13/3,3)#

Jayden Jackson · Answer

undefined To find the centroid of the triangle after reflecting it across the x-axis, we need to find the coordinates of the new vertices and then calculate the centroid.

Given vertices of the triangle:
$ A(4, 1) $, $ B(6, -8) $, and $ C(3, -2) $.

After reflecting across the x-axis, the y-coordinates of the vertices change signs.

New coordinates of the vertices:
$ A'(4, -1) $, $ B'(6, 8) $, and $ C'(3, 2) $.

To find the centroid of the triangle, we take the average of the x-coordinates and the average of the y-coordinates of the vertices.

New centroid coordinates:
$ \left( \frac{4+6+3}{3}, \frac{-1+8+2}{3} \right) $

Simplify to find the centroid:
$ \left( \frac{13}{3}, \frac{9}{3} \right) $

Thus, the new centroid of the triangle after reflection across the x-axis is $ \left( \frac{13}{3}, 3 \right) $.