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

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Answer 1 · 2018-08-08T00:56:00

#color(purple)("Distance moved by centroid " color(chocolate)(vec(GG') = = sqrt 10 ~~ 3.1623 " units"#

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

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

#G(x,y) = ((4 + 5 + 2)/3, (5 + 3 + 1)/3) = (11/3, 3)#

#A'((x),(y)) = 4 a - 3 d = 4 *((4),(5)) - 3 ((4),(2)) = ((4),(14))#

#B'((x),(y)) = 4 b - 3 d = 4 *((5),(3)) - 3 ((4),(2))) = ((8),(6))#

#C'((x),(y)) = 4 c - 3 d = 4 *((2),(1)) - 3 ((4),(2)) = ((-4),(-2))#

#"New Centroid " G'(x,y) = ((4 + 8 - 4)/3, (14 + 6 - 2 )/3) = (8/3, 6)#

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

#color(chocolate)(vec(GG') = sqrt((11/3- 8/3)^2 + (3 - 6)^2) = sqrt 10 ~~ 3.1623 " units"#