Question

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?

Answer 1

`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)]`