A line segment has endpoints at (5 , 8) and (6 , 1). If the line segment is rotated about the origin by pi/2 , translated vertically by -2, and reflected about the y-axis, what will the line segment's new endpoints be?

2 Answers
May 21, 2018

Rule for Rotation by pi/2 or 90 degrees Counter-Clockwise:
(x, y) -> (-y, x)

Rule for Translated Vertically by -2
(x, y) -> (x, y-2)

Rule for Reflection across y-axis
(x, y) -> (-x, y)

In that same order:
(5, 8) -> (-8, 5) -> (-8, 3) -> (8, 3)
(6, 1) -> (-1, 6) -> (-1, 4) -> (1, 4)

To Check, each of these transformations are isometric which means the distance between the points will not change.

Original Points
sqrt((5-6)^2 + (8-1)^2) = sqrt((-1)^2 + (7)^2) = sqrt(1 + 49) = sqrt (50)

New Points
sqrt((8-1)^2 + (3-4)^2) = sqrt((7)^2 + (-1)^2) = sqrt(49 + 1) = sqrt(50)

They have the same distance so the segment endpoints are correct.

May 21, 2018

(8,3)" and "(1,4)

Explanation:

"since there are 3 transformations to be performed"
"label the endpoints"

A=(5,8)" and "B=(6,1)

color(blue)"first transformation"

"under a rotation about the origin of "pi/2

• " a point "(x,y)to(-y,x)

rArrA(5,8)toA'(-8,5)

rArrB(6,1)toB'(-1,6)

color(blue)"second transformation"

"under a vertical translation "((0),(-2))

• " a point "(x,y)to(x,y-2)

rArrA'(-8,5)toA''(-8,3)

rArrB'(-1,6)toB''(-1,4)

color(blue)"third transformation"

"under a reflection in the y-axis"

• " a point "(x,y)to(-x,y)

rArrA''(-8,3)toA'''(8,3)

rArrB''(-1,4)toB'''(1,4)

"After all 3 transformations"

(5,8)to(8,3)" and "(6,1)to(1,4)