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

Feb 6, 2018

Distance moved by centroid G to G' is

$G G ' = \sqrt{{\left(\frac{14}{3}\right)}^{2} + {2}^{2}} \approx \textcolor{red}{5.08}$ units (rounded to 2 decimals)

#### Explanation:

Given A (4,2), B (2, -3), C (8, -2)

Dilated around point D(-3, -1) by a factor (1/3)

To find the distance moved by the centroid

$\vec{A ' D} = \left(\frac{1}{3}\right) \cdot \left(\vec{A D}\right)$

$a ' = \left(\frac{1}{3}\right) \cdot \left(a - d\right) - d = \left(\frac{a}{3}\right) + \left(\frac{2 d}{3}\right)$

a' = (1/3)((4),(2)) + (2/3)((-3),(-1)) = color(purple)(((-2/3),(0))

Similarly $b ' = \left(\frac{1}{3}\right) \cdot \left(b - d\right) - d = \left(\frac{b}{3}\right) + \left(\frac{2 d}{3}\right)$

b' = (1/3)((2),(-3)) + (2/3)((-3),(-1)) =color(purple)( ((0),(-5/3))

$c ' = \left(\frac{1}{3}\right) \cdot \left(c - d\right) - d = \left(\frac{c}{3}\right) + \left(\frac{2 d}{3}\right)$

c' = (1/3)((8),(-2)) + (2/3)((-3),(-1)) =color(purple)( ((2/3),(-4/3))

Centroid G = ((a_x + b_x + c_x)/3, (a_y + b_y + c_y)/3) = color(green)(((14/3), -1)

Dilated centroid point $G ' = \left(\frac{{a}_{x} ' + {b}_{x} ' + {c}_{x} '}{3} , \frac{{a}_{y} ' + {b}_{y} ' + {c}_{y} '}{3}\right)$

G' = ((-2/3) + 0 + (2/3))/3, (0 + (-5/3) + (-4/3))/3 =color(green)( (0, -3)#

Let's use distance formula to find the distance moved by G to G' after dilation

$\vec{G G '} = \sqrt{{\left({G}_{x} - {G}_{x} '\right)}^{2} + {\left({G}_{y} - {G}_{y} '\right)}^{2}}$

$\implies \sqrt{{\left(\left(\frac{14}{3}\right) - 0\right)}^{2} + {\left(- 1 - \left(- 3\right)\right)}^{2}}$

$G G ' = \sqrt{{\left(\frac{14}{3}\right)}^{2} + {2}^{2}} \approx \textcolor{red}{5.08}$ units (rounded to 2 decimals)