A line segment has endpoints at (1 ,2 ) and (5 , 8 ). If the line segment is rotated about the origin by (3 pi ) /2 , translated vertically by  -3 , and reflected about the y-axis, what will the line segment's new endpoints be?

Aug 5, 2017

$\left(- 2 , - 4\right) \text{ and } \left(- 8 , - 8\right)$

Explanation:

$\text{Since there are 3 transformations to be performed name the }$
$\text{endpoints A(1,2) and B(5,8)}$

$\textcolor{b l u e}{\text{First transformation}}$

$\text{under a rotation about the origin of } \frac{3 \pi}{2}$

• " a point "(x,y)to(y,-x)

$\Rightarrow A \left(1 , 2\right) \to A ' \left(2 , - 1\right)$

$\Rightarrow B \left(5 , 8\right) \to B ' \left(8 , - 5\right)$

$\textcolor{b l u e}{\text{Second transformation}}$

$\text{under a dilatation of } \left(\begin{matrix}0 \\ - 3\end{matrix}\right)$

• " a point "(x,y)to(x,y-3)

$\Rightarrow A ' \left(2 , - 1\right) \to A ' ' \left(2 , - 4\right)$

$\Rightarrow B ' \left(8 , - 5\right) \to B ' ' \left(8 , - 8\right)$

$\textcolor{b l u e}{\text{Third transformation}}$

$\text{under a reflection in the y-axis}$

• " a point "(x,y)to(-x,y)

$\Rightarrow A ' ' \left(2 , - 4\right) \to A ' ' ' \left(- 2 , - 4\right)$

$\Rightarrow B ' ' \left(8 , - 8\right) \to B ' ' ' \left(- 8 , - 8\right)$

$\text{after all 3 transformations}$

$\left(1 , 2\right) \to \left(- 2 , - 4\right) \text{ and } \left(5 , 8\right) \to \left(- 8 , - 8\right)$