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

Jun 8, 2018

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

#### Explanation:

$A \left(1 , 5\right) , b \left(- 1 , - 4\right) , C \left(5 , - 6\right) , D \left(- 2 , - 3\right) \text{dilation factor } 2$

Centroid $G = \frac{{x}_{1} + {x}_{2} + {x}_{3}}{3} , \frac{{y}_{1} + {y}_{2} + {y}_{3}}{3}$

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

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)

$G ' \left(x , y\right) = \left(\frac{4 + 0 + 12}{3} , \frac{13 - 5 - 9}{3}\right) = \left(\frac{16}{3} , - \frac{1}{3}\right)$

$\overline{G G '} = \sqrt{{\left(\frac{16}{3} - \frac{5}{3}\right)}^{2} + {\left(- \frac{1}{3} - \frac{5}{3}\right)}^{2}} = 4.18$ units