+912 
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?
+26367 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:
\(\lambda = -2\)
\(\begin{array}{|rcll|} \hline \vec{A'} &=& (\vec{A}-\vec{C})\cdot \lambda + \vec{C} \quad & | \quad \lambda = -2 \\ \binom{8}{-1} &\overset{?}{=}& \Big(\binom{-1}{-1}-\binom{2}{-1} \Big)\cdot (-2) + \binom{2}{-1} \\ &\overset{?}{=}& \binom{-1-2}{-1-(-1)} \cdot (-2) + \binom{2}{-1} \\ &\overset{?}{=}& \binom{-3}{0} \cdot (-2) + \binom{2}{-1} \\ &\overset{?}{=}& \binom{-3\cdot (-2)}{0\cdot (-2)} + \binom{2}{-1} \\ &\overset{?}{=}& \binom{6}{0} + \binom{2}{-1} \\ &\overset{?}{=}& \binom{6+2}{0-1} \\ &\overset{!}{=}& \binom{8}{-1}~ \checkmark \\ \hline \end{array}\)
\( \lambda = 2\)
\(\begin{array}{|rcll|} \hline \vec{A'} &=& (\vec{A}-\vec{C})\cdot \lambda + \vec{C} \quad & | \quad \lambda = 2 \\ \binom{8}{-1} &\overset{?}{=}& \Big(\binom{-1}{-1}-\binom{2}{-1} \Big)\cdot 2 + \binom{2}{-1} \\ &\overset{?}{=}& \binom{-1-2}{-1-(-1)} \cdot 2 + \binom{2}{-1} \\ &\overset{?}{=}& \binom{-3}{0} \cdot 2 + \binom{2}{-1} \\ &\overset{?}{=}& \binom{-3\cdot 2}{0\cdot 2} + \binom{2}{-1} \\ &\overset{?}{=}& \binom{-6}{0} + \binom{2}{-1} \\ &\overset{?}{=}& \binom{-6+2}{0-1} \\ & \ne & \binom{-4}{-1} \\ \hline \end{array}\)
