A line segment has endpoints at $(3 ,4 )$ and $(2 ,5 )$ . 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?

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Answer 1 · 2016-10-20T16:10:38

$(-1, 4) and (0, 5)$

There is a special shortcut to rotate by $pi$ but I will do that rotation the long way, because it shows how do it for any angle.

The first point rotated by $pi$ :

$r = sqrt(3^2 + 4^2)$

$r = 5$

$theta = tan^-1(4/3) + pi$

$(5cos(tan^-1(4/3) + pi), 5sin(tan^-1(4/3) + pi)) =$

$(-3, -4)$

The second point rotated by $pi$ :

$r = sqrt(2^2 + 5^2)$

$r = sqrt(29)$

$theta = tan^-1(5/2) + pi$

$(sqrt(29)cos(tan^-1(5/2) + pi), 5sin(tan^-1(5/2) + pi)) =$

$(-2, -5)$

Translated horizontally by 2 means add two to the x coordinates:

$(-3, -4) -> (-1, -4)$
$(-2, -5) -> (0, -5)$

Reflected about the x axis means multiply the y coordinates by -1:

$(-1, -4) ->(-1, 4)$
$(0, -5) -> (0,5)$

A line segment has endpoints at (3 ,4 )(3 ,4 ) and (2 ,5 )(2 ,5 ). If the line segment is rotated about the origin by pi pi , translated horizontally by 2 2 , and reflected about the x-axis, what will the line segment's new endpoints be?