# A line segment has endpoints at (1 ,5 ) and (5 ,1 ). 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 22, 2017

$\left(5 , - 5\right) \text{ and } \left(9 , - 1\right)$

#### Explanation:

Since there are 3 transformations to be performed, label the endpoints A(1 ,5) and B(5 ,1) and note the change in them after each transformation.

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

• "a point "(x,y)to(-x,-y)

Hence A(1 ,5) → A'(-1 ,-5) and B(5 ,1) → B'(-5 ,-1)

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

• "a point " (x,y)to(x-4,y)

Hence A'(-1 ,-5) →A''(-5 ,-5) and B'(-5 ,-1) → B''(-9 ,-1)

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

• "a point " (x,y)to(-x,y)

Hence A''(-5 ,-5) → A'''(5 ,-5) and B''(-9 ,-1) → B'''(9 ,-1)

Thus after all 3 transformations.

$\left(1 , 5\right) \to \left(5 , - 5\right) \text{ and } \left(5 , 1\right) \to \left(9 , - 1\right)$