# A line segment has endpoints at (9 ,1 ) and (5 ,3). 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?

May 29, 2016

(-9 ,3) , (-5 ,5)

#### Explanation:

Since there are 3 transformations to be performed let's begin by naming the endpoints so we can follow what happens to them.
A(9 ,1) and B(5 ,3)

(1) Under a rotation about the origin of $\pi$

a point (x ,y) → (-x ,-y)

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

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

a point (x ,y) → (x ,y-2)

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

(3) Under a reflection in the x-axis.

a point (x ,y) → (x ,-y)

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

hence (9 ,1) → (-9 ,3) and B(5 ,3) → (-5 ,5)