# A line segment has endpoints at (6 ,5 ) and (5 ,3). If the line segment is rotated about the origin by pi /2 , translated vertically by 2 , and reflected about the x-axis, what will the line segment's new endpoints be?

Dec 11, 2016

$\left(6 , 5\right) \to \left(- 5 , - 4\right) , \left(5 , 3\right) \to \left(- 3 , - 3\right)$

#### Explanation:

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

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

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

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

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

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

Hence A'(-5 ,6) → A''(-5 ,4) and B'(-3 ,5) → B''(-3 ,3)

Third transformation Under a reflection in the x-axis

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

Hence A''(-5 ,4) → A'''(-5 ,-4) and B''(-3 ,3) → B'''(-3 ,-3)

Thus after all 3 transformations.

$\left(6 , 5\right) \to \left(- 5 , - 4\right) \text{ and } \left(5 , 3\right) \to \left(- 3 , - 3\right)$