A triangle has corners at #(1 ,3 )#, #(9 ,4 )#, and #(2 ,2 )#. If the triangle is dilated by a factor of #2 # about point #(9 ,5 ), how far will its centroid move?

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Answer 1 · 2016-06-22T14:49:07

distance #d=5.3851648" "#units

Compute for the centroid #(x_c, y_c)#

#x_c=(x_1+x_2+x_3)/3=(1+9+2)/3=4#
#y_c=(y_1+y_2+y_3)/3=(3+4+2)/3=3#

Centroid #(x_c, y_c)=(4, 3)#

Factor of 2 about the point #(9, 5)#

Let #(x_c', y_c')# be the new centroid

Solve for #x_c'#

#(x_c'-9)/(x_c-9)=2/1#

#(x_c'-9)/(4-9)=2/1#

#(x_c'-9)/(-5)=2#

#x_c'=-10+9#

#x_c'=-1#

Solve for #y_c'#

#(y_c'-5)/(y_c-5)=2/1#

#(y_c'-5)/(3-5)=2/1#

#(y_c'-5)/(-2)=2#

#y_c'=-4+5#

#y_c'=1#

The new centroid #(x_c', y_c')=(-1, 1)#

Solve for the distance between the two centroids

#d=sqrt((x_c-x_c')^2+(y_c-y_c')^2)#

#d=sqrt((4--1)^2+(3-1)^2)#

#d=sqrt((5)^2+(2)^2)#

#d=sqrt(25+4)#

#d=sqrt(29)#

#d=5.3851648" "#units

KIndly see the drawings of the old and new triangles
enter image source here

God bless....I hope the explanation is useful.