A line segment has endpoints at (3 ,8 ) and (4 ,6). If the line segment is rotated about the origin by (pi )/2 , translated vertically by -6 , and reflected about the y-axis, what will the line segment's new endpoints be?

Jun 24, 2016

$\left(3 , 8\right) \to \left(8 , - 3\right) \text{ and } \left(4 , 6\right) \to \left(6 , - 2\right)$

Explanation:

Since there are 3 transformations to be performed name the endpoints A(3 ,8) and B(4 ,6) so we can 'track' them.

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

a point (x ,y) → (-y ,x)

hence A(3 ,8) → A'(-8 ,3) and B(4 ,6) → B'(-6 ,4)

Second transformation: Under a translation $\left(\begin{matrix}0 \\ - 6\end{matrix}\right)$

a point (x ,y) → (x ,y-6)

hence A'(-8 ,3) → A''(-8 ,-3) and B'(-6 ,4) → B''(-6 ,-2)

Third transformation: Under a reflection in the y-axis

a point (x ,y) → (-x ,y)

hence A''(-8 ,-3) → A'''(8 ,-3) and B''(-6 ,-2) → B'''(6 ,-2)

Thus $\left(3 , 8\right) \to \left(8 , - 3\right) \text{ and } \left(4 , 6\right) \to \left(6 , - 2\right)$