# A line segment has endpoints at (7 ,6 ) and (5 ,3 ). If the line segment is rotated about the origin by  pi , translated horizontally by  - 4 , and reflected about the y-axis, what will the line segment's new endpoints be?

Feb 9, 2017

$\left(11 , - 6\right) \text{ and } \left(9 , - 3\right)$

#### Explanation:

Since there are 3 transformations to be performed here, label the endpoints A(7 ,6) and B(5 ,3)

$\textcolor{b l u e}{\text{First transformation"- "Under a rotation about origin of }} \pi$

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

Hence A(7 ,6) → A'(-7 ,-6) and B(5 ,3) → B'(-5 ,-3)

$\textcolor{b l u e}{\text{Second transformation "-"Under a translation}} \left(\begin{matrix}- 4 \\ 0\end{matrix}\right)$

$\text{a point } \left(x , y\right) \to \left(x - 4 , y\right)$

Hence A'(-7 ,-6) → A''(-11 ,-6) and B'(-5 ,-3) →B''(-9 ,-3)

$\textcolor{b l u e}{\text{Third transformation"-" Under a reflection in the y-axis}}$

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

Hence A''(-11 ,-6) → A'''(11 ,-6) and B''(-9 ,-3) → B'''(9 ,-3)

After all 3 transformations.

$\left(7 , 6\right) \to \left(11 , - 6\right) \text{ and } \left(5 , 3\right) \to \left(9 , - 3\right)$