A triangle has corners at $(- 5, 6)$ $(-5 ,6 )$ , $(2, - 3)$ $(2 ,-3 )$ , and $(8, 9)$ $(8 ,9 )$ . If the triangle is dilated by a factor of $5$ $5$ about point #(-3 ,6 ), how far will its centroid move?

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Answer 1 · 2018-02-20T08:58:53

Centroid will move by $23.409$ $color(brown)(23.409$ units

$A (- 5, 6), B (2, - 3), C (8, 9)$ $A(-5,6), B(2,-3), C(8,9)$ Dilated by 5 around $D (- 3, 6)$ $D(-3,6)$

Centroid G(x,y) = (-5/3, 4)#

$A’ = 5A - 4D = 5((-5),(6)) - 4((-3),(6)) = ((-25),(30)) - ((-12),(16)) =color(blue)( ((-13),(14))$

$B’ = 5B - 4D = 5((2),(-3)) - 4((-3),(6)) = ((10),(-15)) - ((-12),(16)) =color(blue)( ((22),(-31))$

$C’ = 5C - 4D = 5((8),(9)) - 4((-3),(6)) = ((40),(35)) - ((-12),(16)) =color(blue)( ((52),(19))$

x Coordinate of new centroid = $G(x) = (a’ + b’ + c’)_x / 3 = (-13 + 22 + 52)/3 = 61/3$

y Coordinate of new centroid = $G(y) = (a’ + b’ + c’)_y / 3 = (14 - 31 + 19)/3 = 12$

Distance moved by centroid $d = sqrt((61/3 + 5/3)^2 + (12-4)^2) = color(brown)(23.4094$

A triangle has corners at (-5 ,6 )(−5,6)(-5 ,6 ), (2 ,-3 )(2,−3)(2 ,-3 ), and (8 ,9 )(8,9)(8 ,9 ). If the triangle is dilated by a factor of 5 55 about point #(-3 ,6 ), how far will its centroid move?