A line segment has endpoints at (6 ,2 ) and (5 ,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?

Jul 1, 2016

$\left(6 , 2\right) \to \left(2 , 8\right) , \left(5 , 7\right) \to \left(7 , 7\right)$

Explanation:

Since there are 3 transformation to be performed, name the endpoints A(6 ,2) and B(5 ,7) so that we can 'track' the changes after each transformation.

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

a point (x ,y) → (-y ,x)

hence A(6 ,2) → A'(-2 ,6) and B(5 ,7) → B'(-7 ,5)

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

a point (x ,y) → (x ,y+2)

hence A'(-2 ,6) → A''(-2 ,8) and B'(-7 ,5) → B''(-7 ,7)

Third transformation: Under a reflection in the y-axis

a point (x ,y) → (-x ,y)

hence A''(-2 ,8) → A'''(2 ,8) and B''(-7 ,7) → B'''(7 ,7)

Thus $\left(6 , 2\right) \to \left(2 , 8\right) \text{ and } \left(5 , 7\right) \to \left(7 , 7\right)$