# A triangle has corners at (8 ,3 ), (4 ,-6 ), and (-2 ,-4 ). If the triangle is dilated by a factor of 5  about point (1 ,-3 ), how far will its centroid move?

Feb 23, 2018

Distance moved by centroid vec(GG’) = color (green)(15.6205

#### Explanation:

Given : $A \left(8 , 3\right) , B \left(4 , - 6\right) , C \left(- 2 , - 4\right)$ Dilated about $D \left(1 , - 3\right)$ by a factor of 5.

To find the distance moved by centroid.

vec(A’D) = 5 * vec(AD)

a’ - d = 5a - 5d or a’ = 5a - 4d

a’((x),(y)) = 5((8),(3)) - 4 ((1),(-3)) = ((40),(15)) - ((4),(-12)) = ((36),(27))

Similarly,

b’((x),(y)) = 5((4),(-6)) - 4 ((1),(-3)) = ((20),(-30)) - ((4),(-12)) = ((16),(18))

b’((x),(y)) = 5((-2),(-4)) - 4 ((1),(-3)) = ((-10),(-20)) - ((4),(-12)) = ((-6),(-8))

Original centroid $g \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}\frac{8 + 4 - 2}{3} \\ \frac{3 - 6 - 4}{3}\end{matrix}\right) = \left(\begin{matrix}\frac{10}{3} \\ - \frac{7}{3}\end{matrix}\right)$

New centroid after dilation

g’((x),(y)) = (((36 + 16 -6)/3),((27+18-8)/3)) = ((46/3),(37/3))

Distance moved by centroid

vec(GG’) = sqrt((10/3-46/3)^2+ (-7/3 + 37/3)^2) = sqrt244 = color (green)(15.6205#