# A line segment has endpoints at (4 ,9 ) and (2 ,9 ). 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?

Jun 8, 2016

(4 ,9) → (-9 ,4) and (2 ,9) → (-9 ,6)

#### Explanation:

Since there are 3 transformations being performed here I will name the endpoints A (4 ,9) and B (2 ,9) so that we can follow what happens to them after each transformation.

First transformation : Rotation about the origin of $\frac{\pi}{2}$

a point (x ,y) → (-y ,x)

hence A(4 ,9) → A'(-9 ,4) and B(2 ,9) → B'(-9 ,2)

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

a point (x ,y) → (x ,y-8)

hence A'(-9 ,4) → A'' (-9 ,-4) and B'(-9 ,2) → B''(-9 ,-6)

Third transformation : Under a reflection in x-axis

a point (x ,y) → (x ,-y)

hence A''(-9 ,-4) → A'''(-9 ,4) and B''(-9 ,-6) → B'''(-9 ,6)

Thus (4 ,9) → (-9 ,4) and (2 ,9) → (-9 ,6)