A triangle has corners at #(-2 ,3 )#, #(3 ,2 )#, and #(5 ,-6 )#. If the triangle is dilated by a factor of #2 # about point #(1 ,-8 ), how far will its centroid move?

Question

A triangle has corners at #(-2  ,3    )#, #(3  ,2   )#, and #(5  ,-6    )#.  If the triangle is dilated by a factor of #2 # about point #(1  ,-8     ), how far will its centroid move?

Claire Akin · Accepted Answer

#d=7.73" (shown in figure)"#

#"The original triangle and its centroid is shown in figure below."#

#"the original centroid can be calculated using:"#

#x=(-2+3+5)/3=6/3=2#

#y=(3+2+-6)/3=-1/3=-0,33#

#E(2,-0.33)#

#"Now dilate A(-2,3) by factor 2 with respect to D(1,-8)"#

#A(-2,3) rArr A'(1-3*2,3+11*2)#

#A'(1-3*2,-8+11*2)#

#A'(-5,14)" (shown in figure below)"#

#![enter image source here](https://tutor.hix.ai) #

#"Dilate B(3,2) by factor 2 with respect to D(1,-8)"#

#B(3,2) rArr B'(1+2*2,-8+10*2)#

#B'(1+2*2,-8+10*2)#

#B'(5,12)" (shown in figure below)"#

#"Dilate C(5,-6) by factor 2 with respect to D(1,-8)"#

#C(5,-6) rArr C'(1+4*2,-8+2*2)#

#C'(1+4*2,-8+2*2)#

#C'(9,-4)" (shown figure below)"#

#"Finally.."#

#"the dilated centroid can be calculated " #

#x'=(-5+5+9)/3=9/3=3#

#y'=(14+12-4)/3#

#y'=22/3=7,33#

#F(3,7.33)#

#"distance between E and F"#

#d=sqrt((3-2)^2+(7.33+0.33)^2)#

#d=sqrt(1+(7.66)^2)#

#d=sqrt(1+58.68)#

#d=sqrt(59.68)#

#d=7.73#

Noah Atkinson · Answer

undefined The centroid of a triangle moves by the same factor as the dilation. Since the triangle is dilated by a factor of 2, the centroid will move twice the distance from its original position to the center of dilation. Therefore, the distance the centroid will move is $2 \times \text{distance from centroid to center of dilation}$. Calculate the distance between the centroid of the original triangle and the center of dilation using the distance formula. Then, multiply this distance by 2 to find how far the centroid will move.