# What is minimum radius of circle given chord length?

What is minimum radius of circle given chord length (Fig.)

I started by setup two line perpendicular to each other.

And find out some area of triangle , then find radius of circle from triangle inscribe in circle formula. If you don't know anything else about the two line segments $AB$, $AC$, and $CD$ except their lengths, then the minimum radius will be half the length of $AB$ (which at its longest is a diameter of the circle).
So, $(1/2)(2+5) = 7/2.$
• We need to be a little careful here. Extending $\overline{CD}$ could possibly lead to a longer chord of the circle, and thus a larger minimum radius. Let's check: If the extension has length $x$, then the Power of a Point theorem says that $3x = 2\cdot 5 = 10$, so that $x = 10/3$. Thus, the extended chord has length $3+10/3 = 19/3$, which is less than $7$, so your calculation works. (On the other hand, if we had, for instance, $|CD| = 1$, then its extended form would have had length $11$.)