# A line segment has endpoints at (3 , 1) and (2 ,3). If the line segment is rotated about the origin by (3pi)/2 , translated horizontally by 7, and reflected about the x-axis, what will the line segment's new endpoints be?

May 4, 2017

$\left(8 , 3\right) \text{ and } \left(10 , 2\right)$

#### Explanation:

$\text{label the endpoints " A(3,1)" and } B \left(2 , 3\right)$

$\textcolor{b l u e}{\text{First transformation "" under a rotation about O of }} \frac{3 \pi}{2}$

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

$\Rightarrow A \left(3 , 1\right) \to A ' \left(1 , - 3\right) \text{ and } B \left(2 , 3\right) \to B ' \left(3 , - 2\right)$

$\textcolor{b l u e}{\text{Second transformation"" under a translation }} \left(\begin{matrix}7 \\ 0\end{matrix}\right)$

$\text{a point " (x,y)to(x+7,y)" hence}$

$A ' \left(1 , - 3\right) \to A ' ' \left(8 , - 3\right) , B ' \left(3 , - 2\right) \to B ' ' \left(10 , - 2\right)$

$\textcolor{b l u e}{\text{Third transformation"" reflection in x-axis}}$

$\text{a point " (x,y)to(x,-y)" hence}$

$A ' ' \left(8 , - 3\right) \to A ' ' ' \left(8 , 3\right) , B ' ' \left(10 , - 2\right) \to B ' ' ' \left(10 , 2\right)$

$\text{After all 3 transformations}$

$\left(3 , 1\right) \to \left(8 , 3\right) \text{ and } \left(2 , 3\right) \to \left(10 , 2\right)$