# A line segment has endpoints at (6 ,5 ) and (2 ,7 ). 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?

Jan 30, 2016

(-4, 7); (0, 7)

#### Explanation:

Use the Rotation, Translation and Reflection mattrix. But this a special case so you can do it in your head.
$\vec{V '} = {M}_{R \left(\theta\right)} \vec{V}$
$\left(- {x}_{1} , - {y}_{1}\right) = {M}_{R \left(\theta\right)} \left({x}_{1} , {y}_{1}\right)$
(-6, -7); (-2,-7)

$\vec{V ' '} = {M}_{T \left(2\right)} \vec{V '}$
$\left(- {x}_{1} + 2 , - {y}_{1}\right) = {M}_{T \left(2\right)} \left(- {x}_{1} , - {y}_{1}\right)$
(-4, -7); (0,-7)

vec(V') = M_(RF(x) vec(V)
$\left(- {x}_{1} + 2 , {y}_{1}\right) = {M}_{R F \left(x\right)} \left(- {x}_{1} + 2 , - {y}_{1}\right)$
(-4, 7); (0, 7)