# A line segment has endpoints at (6 ,5 ) and (2 ,5 ). If the line segment is rotated about the origin by  pi , translated horizontally by  2 , and reflected about the x-axis, what will the line segment's new endpoints be?

Dec 3, 2016

$\left(6 , 5\right) \to \left(- 4 , 5\right) \text{ and } \left(2 , 5\right) \to \left(0 , 5\right)$

#### Explanation:

Since there are 3 transformations to be performed, label the endpoints A(6 ,5) and B(2 ,5)

First transformation Under a rotation about the origin of $\pi$

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

Hence A(6 ,5) → A'(-6 ,-5) and B(2 ,5) → B'(-2 ,-5)

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

$\text{a point } \left(x , y\right) \to \left(x + 2 , y\right)$

Hence A'(-6 ,-5) → A''(-4 ,-5) and B'(-2 ,-5) → B''(0,-5)

Third transformation Under a reflection in the x-axis

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

Hence A''(-4 ,-5) → A'''(-4 ,5) and B''(0 ,-5) → B'''(0 ,5)

Thus after all 3 transformations.

$\left(6 , 5\right) \to \left(- 4 , 5\right) \text{ and } \left(2 , 5\right) \to \left(0 , 5\right)$