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What is the scale factor used to create the dilation?

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The smaller triangle is a pre-image of the bigger triangle. The center of dilation is (2, −1).

What is the scale factor used to create the dilation?

AngelRay  Nov 6, 2017
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3+0 Answers

#2
+8826
+2

I think it's -2

Omi67  Nov 7, 2017
#3
+18715
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What is the scale factor used to create the dilation?
The smaller triangle is a pre-image of the bigger triangle. The center of dilation is (2, -1).
What is the scale factor used to create the dilation?

Let $$\vec{A} = \binom{-1}{-1}$$ before dilation
Let $$\vec{A'} = \binom{8}{-1}$$ after dilation
Let $$\vec{C} = \binom{2}{-1}$$ the center of dilation
Let $$\lambda$$ is the scale factor used to create the dilation

Formula for dilation with vector A:
$$\lambda = -2$$

$$\begin{array}{|rcll|} \hline \vec{A'} &=& (\vec{A}-\vec{C})\cdot \lambda + \vec{C} \quad & | \quad \lambda = -2 \\ \binom{8}{-1} &\overset{?}{=}& \Big(\binom{-1}{-1}-\binom{2}{-1} \Big)\cdot (-2) + \binom{2}{-1} \\ &\overset{?}{=}& \binom{-1-2}{-1-(-1)} \cdot (-2) + \binom{2}{-1} \\ &\overset{?}{=}& \binom{-3}{0} \cdot (-2) + \binom{2}{-1} \\ &\overset{?}{=}& \binom{-3\cdot (-2)}{0\cdot (-2)} + \binom{2}{-1} \\ &\overset{?}{=}& \binom{6}{0} + \binom{2}{-1} \\ &\overset{?}{=}& \binom{6+2}{0-1} \\ &\overset{!}{=}& \binom{8}{-1}~ \checkmark \\ \hline \end{array}$$

$$\lambda = 2$$
$$\begin{array}{|rcll|} \hline \vec{A'} &=& (\vec{A}-\vec{C})\cdot \lambda + \vec{C} \quad & | \quad \lambda = 2 \\ \binom{8}{-1} &\overset{?}{=}& \Big(\binom{-1}{-1}-\binom{2}{-1} \Big)\cdot 2 + \binom{2}{-1} \\ &\overset{?}{=}& \binom{-1-2}{-1-(-1)} \cdot 2 + \binom{2}{-1} \\ &\overset{?}{=}& \binom{-3}{0} \cdot 2 + \binom{2}{-1} \\ &\overset{?}{=}& \binom{-3\cdot 2}{0\cdot 2} + \binom{2}{-1} \\ &\overset{?}{=}& \binom{-6}{0} + \binom{2}{-1} \\ &\overset{?}{=}& \binom{-6+2}{0-1} \\ & \ne & \binom{-4}{-1} \\ \hline \end{array}$$

heureka  Nov 7, 2017
edited by heureka  Nov 7, 2017
#4
+78756
+1

Thanks, Omi and heureka.........I was unsure about this one  !!!

CPhill  Nov 7, 2017
edited by CPhill  Nov 7, 2017

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