isA'B perpendicular to AB'?explain your answer.

the coordinates of the points A and B are (-4,5) and (-5,-4) respectively.A' is reflection image of A with respect to y-axis.B is rotated anticlockwise about the origin O through 90^。     to B'

isA'B perpendicular to AB'?explain your answer.

$$\small{\text{
The vector dot product $\vec{A'B} *\vec{AB'}=0$, if $\vec{A'B}$ perpendicular $\vec{AB'}$ .
}}\\$
\small{\text{
We calculate: $\vec{A'B} = \vec{A'}-\vec{B}=\left(\begin{array}{c}4\\5\end{array}\right)-\left(\begin{array}{c}-5\\-4\end{array}\right)=\left(\begin{array}{c}4-(-5)\\5-(-4)\end{array}\right)}=\left(\begin{array}{c}9\\9\end{array}\right)$
}}$\\$
\small{\text{
and calculate: $\vec{AB'} = \vec{A}-\vec{B'}=\left(\begin{array}{c}-4\\5\end{array}\right)-\left(\begin{array}{c}4\\-5\end{array}\right)=\left(\begin{array}{c}-4-4)\\5-(-5)\end{array}\right)}=
\left(\begin{array}{c}-8\\10\end{array}\right)
$
}}$\\$
\small{\text{
$\vec{A'B} *\vec{AB'} =\left(\begin{array}{c}9\\9\end{array}\right)
*\left(\begin{array}{c}-8\\10\end{array}\right) =9*(-8)+9*10=-72+90=18
$
}}$\\$
\small{\text{
$18 \ne 0 $ so $ \vec{A'B} $ not perpendicular to $ \vec{AB'}$
}}$$

Since A = (-4,5) and A' is the reflection image wrt y-axis, A' = (4, 5).

Since B = (-5,-4) and B' is the 90° anticlockwise rotation, B' = (4,-5)

To determine whether or not A'B is perpendicular to AB', find the slopes of A'B and AB'. If they are negative reciprocals, then the lines will be perpendicular.

Is this enough help?