Problem statement:
$AB$ is the diameter of circle $O$ with a radius of $12$. $P$ is a point on $AB$ between the center point $O$ and $B$ such that $PB = 8$. Find
a) the length of the shortest chord through point $P$.
I would find the answer to this as the shortest cord goes through point $P$ perpendicular to diameter $AB$. So, this question is okay.
b) If the endpoints of the shortest cord in (a) are $X$ and $Y$ find the length of the tangent segment from either $X$ or $Y$ to the vertex of the circumscribed angle $Z$.
It appears to me that there isn't sufficient data to solve this problem b). Can someone throw light how to solve this?
Best Answer
The triangle $ZOX$ has a right angle at $X$ (tangent segment). The triangle $XPO$ is also right angle triangle with $\angle P=90^\circ$ (from (a)). In the $\triangle XPO$ you know all the lengths, so you can write the tanglent of $\angle BOX$ as $$\tan\angle BOX=\frac{XP}{PO}$$ You also have that $$\tan\angle ZOX=\tan\angle BOX=\frac{XZ}{XO}$$ Can you find $XZ$ from here?