# A line segment has endpoints at (4 ,1 ) and (5 ,6). If the line segment is rotated about the origin by pi , translated vertically by -2 , and reflected about the x-axis, what will the line segment's new endpoints be?

Jun 11, 2016

(4 ,1) → (-4 ,3) , (5 ,6) → (-5 ,8)

#### Explanation:

Since there are 3 transformations I am naming the endpoints A(4 ,1) and B(5 ,6) so that we can follow what happens to them after each transformation.

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

a point (x ,y) → (-x ,-y)

hence A(4 ,1) → A'(-4 ,-1) and B(5 ,6) → B'(-5 ,-6)

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

a point (x ,y) → (x ,y-2)

hence A'(-4 ,-1) → A'' (-4 ,-3) and B'(-5 ,-6) → B''(-5 ,-8)

Third transformation: Under a reflection in the x-axis

a point (x ,y) → (x ,-y)

hence A''(-4 ,-3) → A'''(-4 ,3) and B''(-5 ,-8) → B'''(-5 ,8)

Thus (4 ,1) → (-4 ,3) and (5 ,6) → (-5 ,8)