A triangle has corners at $(1, 2)$ $(1 ,2 )$ , $(- 3, - 1)$ $(-3 ,-1 )$ , and $(4, - 2)$ $(4 ,-2 )$ . If the triangle is dilated by a factor of $2$ $2$ about point #(-5 ,7 ), how far will its centroid move?

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Answer 1 · 2018-06-07T10:57:49

$Centroid will move by 8.25 units$ $color(crimson)("Centroid will move by " 8.25 " units"$

$A (1, 2), B (- 3, - 1), C (4, - 2)$ $A (1,2), B (-3,-1), C (4,-2)$ , dilated around $D (- 5, 7)$ $D(-5,7)$ by a factor of 2.

$bar(A'D) = 2 * bar(AD)$

$A' - d = 2 * (a - d)$

$A' = 2a - d$

$A'((x),(y)) = 2 ((1),(2)) - ((-5),(7)) = ((7),(-3))$

Similarly,

$B' = 2b - d$

$B'((x),(y)) = 2 ((-3),(1)) - ((-5),(7)) = ((-1),(-5))$

$C' = 2c - d$

$C'((x),(y)) = 2 ((4),(-2)) - ((-5),(7)) = ((13),(-11))$

$"Centroid of Traingle ABC, " G= ((x_a + x_b + x_c)/3, (y_a + y_b + y_c) /3)$

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

$"Centroid of Traingle A'B'C', " G'= ((x_a' + x_b' + x_c')/3, (y_a' + y_b' + y_c') /3)$

$G'(x,y) = ((7-1+13)/3, (-3-5-11)/3) = (19/3, -19/3)$

$vec(GG') ' gives the distance moved by the centroid"$

$vec(GG') = sqrt((2/3 - 19/3)^2 + (-1/3 + 19/3)^2) = 8.25$

A triangle has corners at (1 ,2 )(1,2)(1 ,2 ), (-3 ,-1 )(−3,−1)(-3 ,-1 ), and (4 ,-2 )(4,−2)(4 ,-2 ). If the triangle is dilated by a factor of 2 22 about point #(-5 ,7 ), how far will its centroid move?