# A line segment has endpoints at (7 ,2 ) and (2 ,3 ). If the line segment is rotated about the origin by  pi , translated horizontally by  - 4 , and reflected about the y-axis, what will the line segment's new endpoints be?

Aug 25, 2016

$\left(7 , 2\right) \to \left(11 , - 2\right) \text{ and } \left(2 , 3\right) \to \left(6 , - 3\right)$

#### Explanation:

Since there are 3 transformations to be performed here, name the endpoints A(7 ,2) and B(2 ,3) so that we can 'track' the coordinates after each transformation.

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

a point (x ,y) → (-x ,-y)

hence A(7 ,2) → A'(-7 ,-2) and B(2 ,3) → B'(-2 ,-3)

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

a point (x ,y) → (x-4 ,y)

hence A'(-7 ,-2) → A''(-11 ,-2) and B'(-2 ,-3) → B''(-6 ,-3)

Third transformation Under a reflection in the y-axis

a point (x ,y) → (-x ,y)

hence A''(-11 ,-2) → A'''(11 ,-2) and B''(-6 ,-3) → B'''(6 ,-3)

Thus after all 3 transformations.

$\left(7 , 2\right) \to \left(11 , - 2\right) \text{ and } \left(2 , 3\right) \to \left(6 , - 3\right)$