A line segment has endpoints at (5 ,9 ) and (8 ,7 ). If the line segment is rotated about the origin by ( pi)/2 , translated vertically by -2 , and reflected about the y-axis, what will the line segment's new endpoints be?

Aug 20, 2016

$\left(5 , 9\right) \to \left(9 , 3\right) \text{ and } \left(8 , 7\right) \to \left(7 , 6\right)$

Explanation:

Since there are 3 transformations to be performed here, name the endpoints A(5 , 9) and B(8 ,7) so that we can 'track' the points after each transformation.

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

a point (x ,y) → (-y ,x)

hence A(5 ,9) → A'(-9 ,5) and B(8 ,7) → B'(-7 ,8)

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

a point (x ,y) → (x , y-2)

hence A'(-9 ,5) → A''(-9 ,3) and B'(-7 ,8) → B''(-7 ,6)

Third transformation Under a reflection in the y-axis

a point (x ,y) → (-x ,y)

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

Thus after all 3 transformations.

$\left(5 , 9\right) \to \left(9 , 3\right) \text{ and } \left(8 , 7\right) \to \left(7 , 6\right)$