A triangle has corners at #(-2 ,1 )#, #(8 ,-5 )#, and #(-1 ,4 )#. If the triangle is dilated by a factor of #5 # about point #(4 ,-6 ), how far will its centroid move?

Question

A triangle has corners at #(-2  ,1    )#, #(8  ,-5   )#, and #(-1  ,4    )#.  If the triangle is dilated by a factor of #5 # about point #(4  ,-6    ), how far will its centroid move?

Aurora Ayala · Accepted Answer

#color(crimson)("Centroid will by 25.75 units"# #A (-2,1), b(8,-5), C(-1,4), D(4,-6) "dilation factor " 5# Centroid #G = (x_1 + x_2 + x_3) /3, (y_1 + y_2 + y_3) / 3# #G(x, y) = ((-2 + 8 - 1) / 3, (1 - 5 + 4) / 3) = (5/3, 0)# #A' ((x),(y)) = 5a - 4d = 5*((-2),(1)) - 4*((4),(-6)) = ((-26), (29))# #B' ((x),(y)) = 5b - 4d = 5*((8),(-5)) - 4*((4),(-6)) = ((24), (-1))# #C' ((x),(y)) = 5c - 4d = 5*((-1),(4)) - 4*((4),(-6)) = ((-21), (44))# #G'(x,y) = ((-26 + 24 -21) / 3, (29 - 1 + 44)/3) = (-23/3, 24)# #bar(GG') = sqrt((-23/3-5/3)^2 + 24^2) = 25.75# units

Parker Allen · Answer

undefined The centroid of a triangle is the point of intersection of its medians. The medians of a triangle are lines drawn from each vertex to the midpoint of the opposite side.

The coordinates of the centroid of a triangle can be found by averaging the coordinates of its vertices.

To find the new coordinates of the centroid after dilation by a factor of 5 about the point (4, -6), first find the coordinates of the original centroid using the given vertices, then apply the dilation transformation to these coordinates.

Original centroid coordinates:
((-2 + 8 - 1)/3, (1 - 5 + 4)/3)

Apply the dilation transformation to each coordinate:

New x-coordinate: (5 * (original x-coordinate - center x-coordinate)) + center x-coordinate
New y-coordinate: (5 * (original y-coordinate - center y-coordinate)) + center y-coordinate

Finally, calculate the distance between the original and new centroids.