A triangle has corners at #(3, 8 )#, ( 5, -2)#, and #( 1, -1)#. If the triangle is reflected across the x-axis, what will its new centroid be?

Question

A triangle has corners at #(3, 8  )#, ( 5,   -2)#, and #( 1,  -1)#.  If the triangle is reflected across the x-axis, what will its new centroid be?

Avery Allen · Accepted Answer

Coordinates of centroid will be #(3,-5/3)# The centroid of a triangle whose vertices are #(x_1,y_1)#, (x_2,y_2)# and (x_3,y_3)# is given by #((x_1+x_2+x_3)/3,(y_1+y_2+y_3)/3)# Hence centrid of given triangle is #((3+5+1)/3,(8+(-2)+(-1))/3)# or #(3,5/3)#. Now, when a point #(x,y)# is reflected across the #x#-axis, its coordinates become #(x,-y)# and hence New coordinates are #(3,-8)#, #(5,2)# and #(1,1)# and even coordinates of centroid will have same rule. Coordinates of centroid will be #(3,-5/3)#

Caroline Aucoin · Answer

undefined The centroid of a triangle is the point where the three medians of the triangle intersect. When a triangle is reflected across the x-axis, the x-coordinates of its vertices remain the same, but the y-coordinates change sign. Therefore, to find the new centroid after reflecting the triangle across the x-axis, we need to take the average of the x-coordinates of the original vertices and the average of the y-coordinates of the reflected vertices.

Original vertices: (3, 8), (5, -2), (1, -1)
Reflected vertices: (3, -8), (5, 2), (1, 1)

Average x-coordinate: (3 + 5 + 1) / 3 = 3
Average y-coordinate: (-8 + 2 + 1) / 3 = -5/3

Therefore, the new centroid of the reflected triangle is (3, -5/3).