# A line segment has endpoints at (8 ,2 ) and (2 ,3 ). If the line segment is rotated about the origin by  pi , translated horizontally by  - 1 , and reflected about the y-axis, what will the line segment's new endpoints be?

Jun 23, 2016

$\left(8 , 2\right) \to \left(9 , - 2\right) \text{ and } \left(2 , 3\right) \to \left(3 , - 3\right)$

#### Explanation:

Since there are 3 transformations to be performed on the points , name them A(8 ,2) and B(2 ,3) so we can 'track' them.

First transformation: Under a rotation about the origin of $\pi$

a point (x ,y) → (-x ,-y)

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

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

a point (x ,y) → (x-1 ,y)

hence A'(-8 ,-2) → A''(-9 ,-2) and B'(-2 ,-3) → B''(-3 ,-3)

Third transformation: Under a reflection in the y-axis

a point (x ,y) → (-x ,y)

hence A''(-9 ,-2) → A'''(9 ,-2) and B''(-3 ,-3) → B'''(3 ,-3)

Thus (8 ,2) → (9 ,-2) and (2 ,3) → (3 ,-3)