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

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Answer 1 · 2018-04-30T11:42:19

Distance moved by centroid $d = 12.67 units$ $d = color(brown)(12.67 " units"$

$A (- 1, 7), B (3, - 5), C (7, - 3)$ $A(-1,7), B(3,-5), C(7,-3)$ Dilated by 5 around $D (- 2, 4)$ $D(-2,4)$

Centroid G(x,y) = (-1+3+7)/3, (7-5-3)/3 = (1, -1/3)#

$A’ = 5A - 4D = 5((-1),(7)) - 4((1),(-1/3)) = ((-5),(35)) - ((4),(-4/3)) =color(blue)( ((-9),(109/3))$

$B’ = 5B - 4D = 5((3),(-5)) - 4((1),(-1/3)) = ((15),(-25)) - ((4),(-4/3)) =color(blue)( ((11),(-71/3))$

$C’ = 5C - 4D = 5((7),(-3)) - 4((1),(-1/3)) = ((35),(-15)) - ((4),(-4/3)) =color(blue)( ((39),(-41/3))$

x Coordinate of new centroid = $G(x) = (a’ + b’ + c’)_x / 3 = (-9 + 11 + 39)/3 = 41/3$

y Coordinate of new centroid = $G(y) = (a’ + b’ + c’)_y / 3 = (109/3 - 71/3 - 41/3)/3 = -1/3$

Distance moved by centroid $d = sqrt((41/3 - 1)^2 + (-1/3 + 1/3)^2) = color(brown)(12.67 " units"$

A triangle has corners at (-1 ,7 )(−1,7)(-1 ,7 ), (3 ,-5 )(3,−5)(3 ,-5 ), and (7 ,-3 )(7,−3)(7 ,-3 ). If the triangle is dilated by a factor of 5 55 about point #(-2 ,4 ), how far will its centroid move?