A line segment has endpoints at (7 ,1 ) and (7 ,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

$\left(7 , 1\right) \to \left(- 9 , 1\right) , \left(7 , 5\right) \to \left(- 9 , 5\right)$

Explanation:

Since there are 3 transformations to be performed, label the endpoints A(7 ,1) and B(7 ,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(7 ,1) → A'(-7 ,-1) and B(7 ,5) → B'(-7 ,-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'(-7 ,-1) → A''(-9 ,-1) and B'(-7 ,-5) → B''(-9 ,-5)

Third transformation Under a reflection in the x-axis

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

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

Thus after all 3 transformations.

$\left(7 , 1\right) \to \left(- 9 , 1\right) \text{ and } \left(7 , 5\right) \to \left(- 9 , 5\right)$