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

Jun 13, 2016

(5 ,2) → (-2 ,-1) and (3 ,1) → (-1 ,1)

#### Explanation:

Since there are 3 transformations to be performed here , name the endpoints A(5 ,2) and B(3 ,1) so we can follow how each point is altered after each transformation.

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

a point (x ,y) → (y ,-x)

hence A(5 ,2) → A'(2 ,-5) and B(3 ,1) → B'(1 ,-3)

Second transformation: Under a translation of $\left(\begin{matrix}0 \\ 4\end{matrix}\right)$

a point (x ,y) → (x ,y+4)

hence A'(2 ,-5) → A''(2 ,-1) and B'(1 ,-3) → B''(1 ,1)

Third transformation: Under a reflection in y-axis

a point (x ,y) → (-x ,y)

hence A''(2 ,-1) → A'''(-2 ,-1) and B''(1 ,1) → B'''(-1 ,1)

Thus (5 ,2) → (-2 ,-1) and (3 ,1) → (-1 ,1)