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

Oct 10, 2016

The distance moved is $4 \sqrt{26}$

#### Explanation:

Let's begin by computing the current centroid, point ${O}_{1}$,

${O}_{1 x} = \frac{6 + 5 - 5}{3} = 2$

${O}_{1 y} = \frac{2 - 8 + 3}{3} = - 1$

Point ${O}_{1} = \left(2 , - 1\right)$

1. Scale point $\left(6 , 2\right)$:

Compute the vector from point $\left(7 , - 2\right)$ to point $\left(6 , 2\right)$:

$\left(6 - 7\right) \hat{i} + \left(2 - - 2\right) \hat{j} = - \hat{i} + 4 \hat{j}$

Scale by 5:

$- 5 \hat{i} + 20 \hat{j}$

Find the new end point:

$\left(- 5 + 7 , 20 + - 2\right) = \left(2 , 18\right)$

1. Scale point $\left(5 , - 8\right)$:

Compute the vector from point $\left(7 , - 2\right)$ to point $\left(5 , - 8\right)$:

$\left(5 - 7\right) \hat{i} + \left(- 8 - - 2\right) \hat{j} = - 2 \hat{i} - 6 \hat{j}$

Scale by 5:

$- 10 \hat{i} - 30 \hat{j}$

Find the new end point:

$\left(- 10 + 7 , - 30 + - 2\right) = \left(- 3 , - 32\right)$

1. Scale point $\left(- 5 , 3\right)$:

Compute the vector from point $\left(7 , - 2\right)$ to point $\left(- 5 , 3\right)$:

$\left(- 5 - 7\right) \hat{i} + \left(3 - - 2\right) \hat{j} = - 12 \hat{i} + 5 \hat{j}$

Scale by 5:

$- 60 \hat{i} + 25 \hat{j}$

Find the new end point:

$\left(- 60 + 7 , 25 + - 2\right) = \left(- 53 , 23\right)$

Our scaled triangle has the vertices, $\left(2 , 18\right)$, $\left(- 3 , - 32\right)$, and $\left(- 53 , 23\right)$. The new centroid, ${O}_{2}$, has coordinates:

${O}_{2 x} = \frac{2 - 3 - 53}{3} = - 18$

${O}_{2 y} = \frac{18 - 32 + 23}{3} = 3$

${O}_{2} = \left(- 18 , 3\right)$

Compute the distance, d, between ${O}_{1}$ and ${O}_{2}$:

$d = \sqrt{{\left(- 18 - 2\right)}^{2} + {\left(3 - - 1\right)}^{2}}$

$d = \sqrt{{\left(- 20\right)}^{2} + {4}^{2}}$

$d = \sqrt{{\left(- 20\right)}^{2} + {4}^{2}}$

$d = \sqrt{416}$

$d = 4 \sqrt{26}$