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

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Answer 1 · 2018-02-23T12:40:26

Distance moved by centroid $vec(GG’) = color (green)(15.6205$

Given : $A (8,3), B (4,-6), C (-2,-4)$ Dilated about $D(1,-3)$ by a factor of 5.

To find the distance moved by centroid.

$vec(A’D) = 5 * vec(AD)$

$a’ - d = 5a - 5d$ or $a’ = 5a - 4d$

$a’((x),(y)) = 5((8),(3)) - 4 ((1),(-3)) = ((40),(15)) - ((4),(-12)) = ((36),(27))$

Similarly,

$b’((x),(y)) = 5((4),(-6)) - 4 ((1),(-3)) = ((20),(-30)) - ((4),(-12)) = ((16),(18))$

$b’((x),(y)) = 5((-2),(-4)) - 4 ((1),(-3)) = ((-10),(-20)) - ((4),(-12)) = ((-6),(-8))$

Original centroid $g((x),(y)) = (((8+4-2)/3),((3-6-4)/3)) = ((10/3),(-7/3))$

New centroid after dilation

$g’((x),(y)) = (((36 + 16 -6)/3),((27+18-8)/3)) = ((46/3),(37/3))$

Distance moved by centroid

$vec(GG’) = sqrt((10/3-46/3)^2+ (-7/3 + 37/3)^2) = sqrt244 = color (green)(15.6205$

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