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

##### 1 Answer
Dec 6, 2016

$d = 7.73 \text{ (shown in figure)}$

#### Explanation:

$\text{The original triangle and its centroid is shown in figure below.}$

$\text{the original centroid can be calculated using:}$

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

$y = \frac{3 + 2 \pm 6}{3} = - \frac{1}{3} = - 0 , 33$

$E \left(2 , - 0.33\right)$

$\text{Now dilate A(-2,3) by factor 2 with respect to D(1,-8)}$

$A \left(- 2 , 3\right) \Rightarrow A ' \left(1 - 3 \cdot 2 , 3 + 11 \cdot 2\right)$

$A ' \left(1 - 3 \cdot 2 , - 8 + 11 \cdot 2\right)$

$A ' \left(- 5 , 14\right) \text{ (shown in figure below)}$

![enter image source here](https://useruploads.socratic.org/v1gFJeqkTVal4HBb8JJA_P2.png) #

$\text{Dilate B(3,2) by factor 2 with respect to D(1,-8)}$

$B \left(3 , 2\right) \Rightarrow B ' \left(1 + 2 \cdot 2 , - 8 + 10 \cdot 2\right)$

$B ' \left(1 + 2 \cdot 2 , - 8 + 10 \cdot 2\right)$

$B ' \left(5 , 12\right) \text{ (shown in figure below)}$

$\text{Dilate C(5,-6) by factor 2 with respect to D(1,-8)}$

$C \left(5 , - 6\right) \Rightarrow C ' \left(1 + 4 \cdot 2 , - 8 + 2 \cdot 2\right)$

$C ' \left(1 + 4 \cdot 2 , - 8 + 2 \cdot 2\right)$

$C ' \left(9 , - 4\right) \text{ (shown figure below)}$

$\text{Finally..}$

$\text{the dilated centroid can be calculated }$

$x ' = \frac{- 5 + 5 + 9}{3} = \frac{9}{3} = 3$

$y ' = \frac{14 + 12 - 4}{3}$

$y ' = \frac{22}{3} = 7 , 33$

$F \left(3 , 7.33\right)$

$\text{distance between E and F}$

$d = \sqrt{{\left(3 - 2\right)}^{2} + {\left(7.33 + 0.33\right)}^{2}}$

$d = \sqrt{1 + {\left(7.66\right)}^{2}}$

$d = \sqrt{1 + 58.68}$

$d = \sqrt{59.68}$

$d = 7.73$