# A line segment has endpoints at (8 ,1 ) and (7 ,5 ). 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?

Apr 12, 2016

(-10 , 1) and (-9 , 5)

#### Explanation:

Let's begin by naming the endpoints A(8 , 1) and B(7 , 5)

Under a rotation of $\pi \text{ about the origin }$
a point ( x , y) → (-x , -y)

step 1 : A (8 , 1 ) → A' (-8 , -1) and B (7 , 5) → B'(-7 , -5)

Under a translation of $\left(\begin{matrix}- 2 \\ 0\end{matrix}\right) \text{ point (x , y) → (x-2 , y)}$

step 2 : A'(-8 , -1) → A''(-10 , -1) and B'(-7 ,-5) → B''(-9 , -5)

Under a reflection in the x-axis, a point (x , y) → (x , -y)

step 3 : A'' (-10 , -1) → A'''(-10 , 1) and B''(-9 , -5) → B'''(-9 , 5)

hence new endpoints are (-10 , 1) and (-9 , 5)