In the figure above, line $\iota$ (not shown) is perpendicular to segment $AB$ and bisects segment $AB$. Which of the following points lies on line $\iota$?

Given line l bisects AB and perpendicular to AB.

To find point lie on the line

Slope of the line perpendicular to AB

${ m }_{ 1 }{ m }_{ 2 }=-1\\ -1m_{ 2 }=-1\\ m_{ 2 }=1$

And it will pass through the midpoint of AB

Let coordinates of midpoint be $(x,y)$

$x=\cfrac { 0+2 }{ 2 } =1\\ y=\cfrac { 0+2 }{ 2 } =1\\ (x,y)=(1,1)$

So equation of line passing through $(1,1)$ and having slope $m=1$

$(y-1)=1\times (x-1)\\ y-1=x-1\\ y=x$

Hence $(3,3)$ be the point that lie on this line.