A line segment has endpoints at #(5 ,9 )# and #(8 ,7 )#. If the line segment is rotated about the origin by #( pi)/2 #, translated vertically by #-8 #, and reflected about the y-axis, what will the line segment's new endpoints be?

1 Answer
Mar 31, 2016

Answer:

#A''' = [(-1, 0), (0, 1)] [(-9 ), (-3)]=[(9), (-3)]#

#B''' = [(-1, 0), (0, 1)] [(-7), (0)]=[(7), (0)]#

Explanation:

Given : Two points #A(5,9)# and #B(8,7)#. Rotate by #pi/2#, translate vertically -8 and reflected about y-axis
Required : New end-point of the segment #A'# and #B'#
Solution Strategy : a) Rotate b) Translated Vertically c) Reflect

a) Rotate using Rotation Matrix :
#R(pi/2) = [(cos(pi/2), -sin(pi/2)), (sin(pi/2), cos(pi/2))]=[(0, -1), (1, 0)]#
#A' = [(0, -1), (1, 0)] [(5 ), (9)]=[(-9), (5)]#

#B' = [(0, -1), (1, 0)] [(8), (7)]=[(-7), (8)]#

b) Translate Vertically :
Translation Operation of vector, #P# by #delta_(x,y)# is given by:
#vec(P') = vecP + vecdelta_(x,y) # where
#delta_(x,y)=[(delta_x), (delta_y)]# thusd
#A'' = [(-9), (5)] + [(0), (-8)]=[(-9), (-3)]#

#B''=[(-7), (8)]+[(0), (-8)]= [(-7), (0)]=#

c) Reflection Vertically :
Reflection Matrix about y, is #Rf_y=[(-1, 0), (0, 1)]#
#A''' = [(-1, 0), (0, 1)] [(-9 ), (-3)]=[(9), (-3)]#

#B''' = [(-1, 0), (0, 1)] [(-7), (0)]=[(7), (0)]#