A triangle as corners at $(4, 5)$ $(4 ,5 )$ , $(5, 3)$ $(5 ,3)$ , and $(2, 1)$ $(2 ,1)$ . If the triangle is dilated by a factor of $4$ $4$ 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 $G(x,y) = ((x_a + x_b + x_c) /3, (y_a + y_b + y_c)/3)$

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

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

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

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

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

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

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

A triangle as corners at (4 ,5 )(4,5)(4 ,5 ), (5 ,3)(5,3)(5 ,3), and (2 ,1)(2,1)(2 ,1). If the triangle is dilated by a factor of 4 44 about #(4 ,2 ), how far will its centroid move?