A triangle has corners at #(6 ,3 )#, #(4 ,-1 )#, and #(2 ,-9 )#. If the triangle is dilated by a factor of #5 # about point #(1 ,-5 ), how far will its centroid move?

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Answer 1 · 2018-07-14T08:28:41

#color(green)("Distance moved by centroid " color(purple)(vec(GG') ~~ 13.4164 " units"#

#A(6,3), B(4,-1), C(2,-9), " about point " D (1,-5), " dilation factor "5#

Centroid #G(x,y) = ((x_a + x_b + x_c) /3, (y_a + y_b + y_c)/3)#

#G(x,y) = ((6+4+2)/3, (3 - 1 - 9)/3) = (4, -7/3)#

#A'((x),(y)) = 5a - 4d = 5*((6),(3)) - 4*((1),(-5)) = ((26),(35))#

#B'((x),(y)) = 5b - 4d = 5*((4),(-1)) - 4*(1),(-5)) = ((16),15))#

#C'((x),(y)) = 5c - 4d = 5*((2),(-9)) - 4*((1),(-5)) = ((6),(-25))#

#"New Centroid " G'(x,y) = ((26+ 16 + 6)/3,(35+ 15 -25)/3) = (16,25/3)#

#color(green)("Distance moved by centroid " #

#color(purple)(vec(GG') = sqrt((4- 16)^2 + (-7/3 + 25/3)) ~~ 13.4164 " units"#