# A triangle has corners at (-7 ,-2 ), (3 ,-1 ), and (5 ,7 ). If the triangle is dilated by a factor of 2/5  about point (6 ,-8 ), how far will its centroid move?

Feb 12, 2018

Centroid has moved by a distance of $\approx \textcolor{g r e e n}{6.55}$ units

#### Explanation:

Given : A (-7,-2), B (3,-1), C(5,7)

Dilated about D(6,-8) and dilation factor 2/5

To find the distance moved by centroid.

Centroid G = (-7+3+5)/3, (-2-1+7)/3 = color(brown)((1/3,4/3)

$\vec{A ' D} = \left(\frac{2}{5}\right) \cdot \vec{A D}$

$a ' - d = \left(\frac{2}{5}\right) \left(a - d\right)$ or $a ' = 0.4 a + 0.6 d$

=> 0.4((-7),(-2)) + 0.6((6),(-8)) = ((-2.8),(-0.8)) + ((3.6,(-4.8)) = ((0.8),(-5.6))

color(blue)(A' (0.8, -5.6)

Similarly,

$\vec{B ' D} = 5 \cdot \vec{B D}$

$b ' - d = \left(\frac{2}{5}\right) \left(b - d\right)$ or $b ' = 0.4 b + 0.6 d$

=> 0.4((3),(-1)) + 0.6((6),(-8)) = ((1.2),(-0.4)) + ((3.6,(-4.8)) = ((4.8),(-5.2))

color(blue)(B' (4.8, -5.2)

$\vec{C ' D} = \left(\frac{2}{5}\right) \cdot \vec{C D}$

$c ' - d = \left(\frac{2}{5}\right) \left(c - d\right)$ or $c ' = 0.4 c + 0.6 d$

=> 0.4((5),(7)) + 0.6((6),(-8)) = ((2),(2.8)) + ((3.6,(-4.8)) = ((5.6),(-2))

color(blue)(C' (5.6, -2)

New centroid G' = (0.8 +4.8+5.6)/3, (-5.6-5.2-2)/3 = color(brown)((11.2/3, -12.8/3)#

Distance moved by centroid is

$\vec{G G '} = \sqrt{{\left(\left(\frac{1}{3}\right) - \frac{11.2}{3}\right)}^{2} + {\left(\left(\frac{4}{3}\right) + \left(\frac{12.8}{3}\right)\right)}^{2}} \approx \textcolor{g r e e n}{6.55}$ units