# A line segment has endpoints at (4 ,0 ) and (2 ,1 ). If the line segment is rotated about the origin by ( pi)/2 , translated vertically by -8 , and reflected about the x-axis, what will the line segment's new endpoints be?

Dec 3, 2016

$\left(4 , 0\right) \to \left(0 , 4\right) \text{ and } \left(2 , 1\right) \to \left(- 1 , 6\right)$

#### Explanation:

Since there are 3 transformations to be performed, label the endpoints A(4 ,0) and B(2 ,1)

First transformation Under a rotation about the origin of $\frac{\pi}{2}$

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

Hence A(4 ,0) → A'(0 ,4) and B(2 ,1) → B'(-1 ,2)

Second transformation Under a translation $\left(\begin{matrix}0 \\ - 8\end{matrix}\right)$

$\text{a point } \left(x , y\right) \to \left(x , y - 8\right)$

Hence A'(0 ,4) → A''(0 ,-4) and B'(-1 ,2) → B''(-1 ,-6)

Third transformation Under a reflection in the x-axis

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

Hence A''(0 ,-4) → A'''(0 ,4) and B''(-1 ,-6) → B'''(-1 ,6)

Thus after all 3 transformations.

$\left(4 , 0\right) \to \left(0 , 4\right) \text{ and } \left(2 , 1\right) \to \left(- 1 , 6\right)$