# A line segment has endpoints at (4 ,2 ) and (5 ,3 ). If the line segment is rotated about the origin by ( 3 pi)/2 , translated horizontally by  2 , and reflected about the x-axis, what will the line segment's new endpoints be?

Jul 28, 2018

$\left(4 , 4\right) \text{ and } \left(5 , 5\right)$

#### Explanation:

$\text{Since there are 3 transformations to be performed label}$
$\text{the endpoints}$

$A = \left(4 , 2\right) \text{ and } B = \left(5 , 3\right)$

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

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

$\text{a point } \left(x , y\right) \to \left(y , - x\right)$

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

$B \left(5 , 3\right) \to B ' \left(3 , - 5\right)$

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

$\text{under a horizontal transformation } \left(\begin{matrix}2 \\ 0\end{matrix}\right)$

• " a point "(x,y)to(x+2,y)

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

$B ' \left(3 , - 5\right) \to B ' ' \left(5 , - 5\right)$

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

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

• " a point "(x,y)to(x,-y)

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

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

$\text{After all 3 transformations}$

$\left(4 , 2\right) \to \left(4 , 4\right) \text{ and } \left(5 , 3\right) \to \left(5 , 5\right)$