A triangle has corners at $(-1 ,3 )$ , $(3 ,-2 )$ , and $(8 ,4 )$ . If the triangle is dilated by a factor of $5$ about point #(-2 ,6 ), how far will its centroid move?

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Answer 1 · 2018-02-05T09:13:35

Centroid moves by a distance of $~~color(green)(24.5)$ units

Given A (-1,3), B(3,-2), C (8,4),

Dilation point D (-2,6), Dilation factor 5

To find
1. Present centroid,
2. centroid after dilation and
3. distance between the two centroids.

Present centroid $G = (-1 + 3 + 8)/3, (3 - 2 + 4) /3 = color(brown)((10/3, 5/3)$

$bar(DA') = 5 * (bar(DA))$

$a' - d = 5a - 5d$

$a' = 5a - 4d = 5((-1),(3)) - 4((-2),(6))$

$=> ((-5),(15)) - ((-8),(24)) = ((3),(-9))$

$color(green)(A' (3, -9)$

$bar(DB') = 5 * bar(DB)$

$b' = 5b - 4d = 5((3),(-2)) - 4 ((-2),(6))$

$=> ((15),(-10)) - ((-8),(24)) = ((23),(-34))$

$color(green)(B'(23, -34)$

Similarly, $bar(DC') = 5 * bar(DC)$

$c' = 5c - 4d = 5((8),(4)) - 4((-2),(6))$

$=> ((40),(20)) - ((-8),(24)) = ((48),(-4))$

$color(green)(C' (48, -4)$

Centroid of dilated triangle A'B'C' is

$G' = (3 + 23 + 48) / 3, (-9-34-4)/3 = color(brown)((74/3, -47/3)$

Distance between centroids can be found using distance formula,

$bar(GG') = sqrt(((74/3)-(10/3))^2 + ((-47/3) -(5/3))^2)$

$bar(GG') = sqrt((64/3)^2 + (52/3)^2) ~~ color(green)(24.5)$
(corrected to one decimal)

A triangle has corners at (-1 ,3 )(-1 ,3 ), (3 ,-2 )(3 ,-2 ), and (8 ,4 )(8 ,4 ). If the triangle is dilated by a factor of 5 5 about point #(-2 ,6 ), how far will its centroid move?