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

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Answer 1 · 2018-06-08T02:47:55

$Centroid has moved by a distance of 4.18 units$ $color(green)("Centroid has moved by a distance of " 4.18 " units"$

$A (1, 5), b (- 1, - 4), C (5, - 6), D (- 2, - 3) dilation factor 2$ $A (1,5), b(-1,-4), C(5,-6), D(-2,-3) "dilation factor " 2$

Centroid $G = \frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3}$ $G = (x_1 + x_2 + x_3) /3, (y_1 + y_2 + y_3) / 3$

$G (x, y) = (\frac{1 - 1 + 5}{3}, \frac{5 - 4 - 6}{3}) = (\frac{5}{3}, \frac{5}{3})$ $G(x, y) = ((1 - 1 + 5) / 3, (5 - 4 - 6) / 3) = (5/3, 5/3)$

$A' ((x),(y)) = 2a - d = 2*((1),(5)) - ((-2),(-3)) = ((4, 13)$

$B' ((x),(y)) = 2b - d = 2*((-1),(-4)) -( (-2),(-3)) = ((0, -5)$

$C' ((x),(y)) = 2c - d = 2*((5),(-6)) - ((-2),(-3)) = ((12, -9)$

$G'(x,y) = ((4 + 0 + 12) / 3, (13 - 5 - 9)/3) = (16/3, -1/3)$

$bar(GG') = sqrt((16/3-5/3)^2 + (-1/3-5/3)^2) = 4.18$ units

A triangle has corners at (1 ,5 )(1,5)(1 ,5 ), (-1 ,-4 )(−1,−4)(-1 ,-4 ), and (5 ,-6 )(5,−6)(5 ,-6 ). If the triangle is dilated by a factor of 2 22 about point (-2 ,-3 )(−2,−3)(-2 ,-3 ), how far will its centroid move?