A line segment has endpoints at (2 ,3 ) and (6 ,5). If the line segment is rotated about the origin by (3pi )/2 , translated horizontally by -1 , and reflected about the x-axis, what will the line segment's new endpoints be?

Oct 3, 2016

$\left(2.3\right) \to \left(2 , 2\right) , \left(6 , 5\right) \to \left(4 , 6\right)$

Explanation:

Since there are 3 transformations to be performed here, name the endpoints A(2 ,3) and B(6 ,5), so that we can follow the changes that occur.

First transformation Under a rotation about the origin of $\frac{3 \pi}{2}$

a point (x ,y) → (y ,-x)

hence A(2 ,3) → A'(3 ,-2) and B(6 ,5) → B'(5 ,-6)

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 ,-2) → A''(2 ,-2) and B'(5 ,-6) → B''(4 ,-6)

Third transformation Under a reflection in the x-axis

a point (x ,y) → (x ,-y)

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

Thus after all 3 transformations.

$\left(2 , 3\right) \to \left(2 , 2\right) \text{ and } \left(6 , 5\right) \to \left(4 , 6\right)$