# A line segment has endpoints at (1 ,4 ) and (3 ,9 ). If the line segment is rotated about the origin by  pi , translated horizontally by  - 2 , and reflected about the x-axis, what will the line segment's new endpoints be?

Jul 25, 2016

$\left(1 , 4\right) \to \left(- 3 , 4\right) \text{ and } \left(3 , 9\right) \to \left(- 5 , 9\right)$

#### Explanation:

Since there are 3 transformations to be performed, name the endpoints A(1 ,4) and B(3 ,9) so we can 'track' them.

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

a point (x ,y) → (-x ,-y)

hence A(1 ,4) → A'(-1 ,-4) and B(3 ,9) → B'(-3 ,-9)

Second transformation Under a translation of$\left(\begin{matrix}- 2 \\ 0\end{matrix}\right)$

a point (x ,y) → (x-2 ,y)

Hence A'(-1 ,-4) → A''(-3 ,-4) and B'(-3 ,-9) → B''(-5 ,-9)

Third transformation Under a reflection in the x-axis

a point (x ,y) → (x ,-y)

hence A''(-3 ,-4) → A'''(-3 ,4) and B''(-5 ,-9) → B'''(-5 ,9)

Thus $\left(1 , 4\right) \to \left(- 3 , 4\right) \text{ and } \left(3 , 9\right) \to \left(- 5 , 9\right)$