A triangle has corners at #(-3 ,5 )#, #(7 ,6 )#, and #(1 ,-4 )#. If the triangle is dilated by a factor of #2/3 # about point #(-3 ,4 ), how far will its centroid move?

Question

A triangle has corners at #(-3  ,5    )#, #(7  ,6   )#, and #(1  ,-4    )#.  If the triangle is dilated by a factor of #2/3 # about point #(-3  ,4     ), how far will its centroid move?

Maverick Smith · Accepted Answer

Thus the new corners of the dilated triangle #A'B'C"# are
#A'(-3, 14/3), B'(11/3, 16/3), C'(-1/3, -4/3)#

For each vertex we need to find the dilation of that point by 2/3 around (-3,4): #A(-3,5), B(7, 6), C(1, -4)# Dilate #B'=D_((-3,4), 2/3) *B' (7,6)=>[7-(-3), 6-4] *2/3 + [-3, 4]= (11/3, 16/3)# #A'=D_((-3,4), 2/3)* A'(-3,5) =>[-3-(-3), 5-4] *2/3 + [-3, 4]= (-3, 14/3)# #C'=D_((-3,4), 2/3) *C'(1,-4) =>[1-(-3), -4-4] *2/3 + [-3, 4]= (-1/3, -4/3)# Thus the new corners of the dilated triangle #A'B'C"# are #A'(-3, 14/3), B'(11/3, 16/3), C'(-1/3, -4/3)#

Sadie Allen · Answer

undefined The centroid of a triangle moves in the same ratio as the dilation factor. Therefore, if the triangle is dilated by a factor of 2/3 about point (-3, 4), the centroid will move by 2/3 of the distance between the original centroid and the center of dilation.

First, calculate the original centroid of the triangle using the coordinates of its vertices.

Centroid = ((-3 + 7 + 1) / 3, (5 + 6 - 4) / 3)
Centroid = (5/3, 7/3)

Next, calculate the distance between the original centroid and the center of dilation (-3, 4).

Distance = sqrt(((-3 - 5/3)^2) + ((4 - 7/3)^2))

Now, multiply this distance by the dilation factor (2/3) to find how far the centroid moves.

Distance moved by the centroid = (2/3) * Distance