# A line segment has endpoints at (9 ,6 ) 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?

Nov 10, 2016

$\left(9 , 6\right) \to \left(6 , 7\right) , \left(5 , 3\right) \to \left(3 , 3\right)$

#### Explanation:

Since there are 3 transformations to be performed here, label the points A(9 ,6) and B(5 ,3) so that the change to each point after each transformation can be noted.

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

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

hence A(9 ,6) →A'(6 ,-9) and B(5 ,3) → B'(3 ,-5)

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

$\text{ a point} \left(x , y\right) \to \left(x + 0 , y + 2\right)$

hence A'(6 ,-9) → A''(6 ,-7) 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''(6 ,-7) → A'''(6 ,7) and B''(3 ,-3) → B'''(3 ,3)

After all 3 transformations.

$\left(9 , 6\right) \to \left(6 , 7\right) \text{ and } \left(5 , 3\right) \to \left(3 , 3\right)$