Question

A line segment has endpoints at `(8 , 4)` and `(1 , 2)`. If the line segment is rotated about the origin by `(3pi)/2`, translated vertically by `4`, and reflected about the x-axis, what will the line segment's new endpoints be?

Answer 1

Summarizing
transformed coordinates are
from
`x_11=8, y_11=4`
`x_21=1, y_21=2`
to
`x_14,y_14=4,4`
`x_24,y_24=2,-3`

Explanation:

`x_11=8, y_11=4`

`x_21=1, y_21=2`

`r_11=sqrt(x_11^2+y_11^2)=sqrt(8^2+4^2)=sqrt(64+16)=sqrt80`

`theta_11=tan^-1(y_11/x_11)=tan^-1(4/2)=26.565^@`

`r_21=sqrt(x_21^2+y_21^2)=sqrt(1^2+2^2)=sqrt(1+4)=sqrt5`

`theta_21=tan^-1(y_21/x_21)=tan^-1(2/1)=63.435^@^@`

Rotation is

`alpha=(3pi)/2=270^@`

`r_12=r_11`

`theta_12=theta_11+270^@`

`r_12=sqrt80`

`theta_12=26.565^@+270^@=296.565^@`

`r_22=r_21`

`theta_22=theta_21+270^@`

`r_22=sqrt5`

`theta_22=63.435^@+270^@=333.435^@`

`(x_12,y_12)=(r_12costheta_12,r_12sintheta_12)`
`=sqrt80cos296.565^@,sqrt80sin296.565^@=-=(4,-8)`

`x_12=4`
`y_12=-8`

`(x_22,y_22)=(r_22costheta_22,r_22sintheta_22)`
`=sqrt5cos333.435^@,sqrt5sin333.435^@=-=(2,-1)`
`x_22=2`
`y_22=-1`

translation is +4

`x_13=x_12`
`y_13=y_12+4`

`x_13=4`
`y_13=-8+4=-4`

`x_23=x_22`
`y_23=y_22+4`

`x_23=2`
`y_23=-1+4=3`

Reflection about x axis

`x_14=x_13`
`y_14=-y_12`

`x_14=4`
`y_14=4`

`x_24=x_23`
`y_23=-y_22`

`x_23=2`
`y_23=-3`

Summarizing

`x_11,y_11=8,4`
`x_21,y_21=1,2`
Rotation by `(3pi)/2`
`x_12,y_12=4,-8`
`x_22,y_22=2,-1`
translated bertically by 4
`x_13,y_13=4,-4`
`x_23,y_23=2,3`
reflected about x axis
`x_14,y_14=4,4`
`x_24,y_24=2,-3`