A line segment has endpoints at (5 ,3 ) and (5 ,4). If the line segment is rotated about the origin by pi /2 , translated horizontally by -1 , and reflected about the x-axis, what will the line segment's new endpoints be?

Oct 2, 2016

$\left(5 , 3\right) \to \left(- 4 , - 5\right) \text{ and } \left(5 , 4\right) \to \left(- 5 , - 5\right)$

Explanation:

Since there are 3 transformations to be performed, name the endpoints A(5 ,3) and B(5 ,4) so that we can follow the changes that occur to them.

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

a point (x , y) → (-y ,x)

hence A(5 ,3) → A'(-3 ,5) and B(5 ,4) → B'(-4 ,5)

Second transformation Under a translation $\left(\begin{matrix}- 1 \\ 0\end{matrix}\right)$

a point (x ,y) → (x-1 ,y+0) → (x-1 ,y)

hence A'(-3 ,5) → A''(-4 ,5) and B'(-4 ,5) → B''(-5 ,5)

Third transformation Under a reflection in the x-axis

a point (x ,y) → (x ,-y)

hence A''(-4 ,5) → A'''(-4 ,-5) and B''(-5 ,5) → B'''(-5 ,-5)

Thus after all 3 transformations.

$\left(5 , 3\right) \to \left(- 4 , - 5\right) \text{ and } \left(5 , 4\right) \to \left(- 5 , - 5\right)$