# A line segment has endpoints at (4 ,1 ) and (5 ,3). If the line segment is rotated about the origin by pi , translated vertically by -2 , and reflected about the x-axis, what will the line segment's new endpoints be?

Apr 14, 2016

(-4 , 3) , (-5 , 5)

#### Explanation:

under a rotation of $\pi$" about the origin " #

a point (x , y) → (-x , -y)

Name the points A(4 , 1 ) and B(5 , 3)
$\textcolor{red}{\text{----------------------------------------------}}$
Step 1 :

A(4 , 1) → A'(-4 ,-1) and B(5 , 3) → B'(-5 , -3)
$\textcolor{red}{\text{----------------------------------------------}}$

Under a translation of $\left(\begin{matrix}0 \\ - 2\end{matrix}\right)$

a point (x , y) → (x , y-2)
$\textcolor{red}{\text{-----------------------------------------------}}$
Step 2 :

A'(-4 , -1) →A'' (-4 , -3) and B'(-5 ,-3) → B''(-5 , -5)
$\textcolor{red}{\text{--------------------------------------------------}}$

Under a reflection in the x-axis

a point (x , y) → (x , -y)
$\textcolor{red}{\text{-------------------------------------------------}}$

Step 3 :

A''(-4 , -3) → A'''(-4 , 3) and B''(-5 , -5) → (-5 , 5)
$\textcolor{red}{\text{----------------------------------------------------}}$