# A line segment has endpoints at (3 ,1 ) and (0 , 0 ). 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?

Jun 25, 2016

(3 ,1) → (0 ,-3) and (0 ,0) → (1 ,0)

#### Explanation:

Since there are 3 transformations to be performed, name the endpoints A(3 ,1) and B(0 ,0) so that 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 ,1) →A' (-1 ,3) and B(0 ,0) → B'(0 ,0)

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

a point (x ,y) → (x+1 ,y)

hence A'(-1 ,3) → A''(0 ,3) and B'(0 ,0) → B''(1 ,0)

Third transformation Under a feflection in the x-axis

a point (x ,y) → (x ,-y)

hence A''(0 ,3) → A'''(0 ,-3) and B''(1 ,0) → B'''(1 ,0)

Thus (3 ,1) → (0 ,-3) and (0 ,0) → (1 ,0)