A line segment has endpoints at $(4 ,0 )$ and $(2 ,9 )$ . If the line segment is rotated about the origin by $( pi)/2$ , translated vertically by $-8$ , and reflected about the x-axis, what will the line segment's new endpoints be?

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Answer 1 · 2016-10-11T04:40:07

The endpoints are $(0, 4)$ and $(-9, 6)$

Rotate $(4, 0) by pi/2$ :

$|4,0| = 4$

$theta = tan^-1(0/4) = 0$

The rotated angle

$theta_r = pi/2$

$(4cos(pi/2), 4sin(pi/2)) = (0, 4)$

Rotate $(2, 9) by pi/2$ :

$|2,9| = sqrt(2^2 + 9^2)$

$|2,9| = sqrt(85)$

$theta = tan^-1(9/2)$

The rotated angle

$theta_r = tan^-1(9/2) + pi/2$

$(sqrt(85)cos(tan^-1(9/2) + pi/2), sqrt(85)sin(tan^-1(9/2) + pi/2)) = (-9, 2)$

Translate both points vertically by -8:
$(0, 4-8) = (0, -4)$
$(-9, 2-8) = (-9, -6)$

Reflect about the x axis:

$(0, -1*-4) = (0, 4)$
$(-9, -1*-6) = (-9, 6)$

A line segment has endpoints at (4 ,0 )(4 ,0 ) and (2 ,9 )(2 ,9 ). If the line segment is rotated about the origin by ( pi)/2 ( pi)/2 , translated vertically by -8 -8 , and reflected about the x-axis, what will the line segment's new endpoints be?