You are given the following: a circle, one of its diameters $AB$ and a point $X$. (a) Using only a straight-edge, show in the given figure how to draw a line perpendicular to $AB$ passing through $X$. No credit will be given without full justification. (Recall that a straight-edge is a ruler without any markings. Given two points, a straight-edge can be used to draw the line passing through the given points.) (b) Do NOT draw any of your work for this part in the given figure. Reconsider your procedure to see if it can be made to work if the point $X$ is in some other position, e.g., when it is inside the circle or to the ``left/right'' of the circle. Clearly specify all positions of the point $X$ for which your procedure in part (a), or a small extension/variation of it, can be used to obtain the perpendicular to $AB$ through $X$. Justify your answer.
You are given the following: a circle, one of its diameters $AB$ and a point $X$.\\
(a) Using only a straight-edge, show in the given figure how to draw a line perpendicular to $AB$ passing through $X$. No credit will be given without full justification. (Recall that a straight-edge is a ruler without any markings. Given two points, a straight-edge can be used to draw the line passing through the given points.)\\
(b) Do NOT draw any of your work for this part in the given figure. Reconsider your procedure to see if it can be made to work if the point $X$ is in some other position, e.g., when it is inside the circle or to the ``left/right'' of the circle. Clearly specify all positions of the point $X$ for which your procedure in part (a), or a small extension/variation of it, can be used to obtain the perpendicular to $AB$ through $X$. Justify your answer.