# Circumscribe a square about a semicircle!

• Feb 2nd 2013, 01:12 PM
Jammix
Circumscribe a square about a semicircle!
Thanks!
• Feb 2nd 2013, 04:01 PM
Soroban
Re: Circumscribe a square about a semicircle!
Hello, Jammix!

I suspect that I know the solution . . . but it's only a guess.

Quote:

Circumscribe a square about a semicircle.

Code:

                R     B * - -.-.*.o.*.-.- * C       |  .*:::::|:::::*.|       | *:::::::|:::::::o S       |*::::::::|:::::* |       |:::::::::|:::* r |       *:::::::::|:*    *     Q o:-:-:-:-:O - - - o       *:::::::* :      *       |:::::*r  :      |       | ::*    :      |     A * o - - - * - - - * D         P      E
Square $ABCD$ is circumscribed about the (tilted) semicircle.
The circle has center $O$ and radius $r\!:\;OP = OQ = OR = OS = r.$

$\Delta OEA$ is an isosceles right triangle.
Hypotenuse $PO = r \quad\Rightarrow\quad OE = \tfrac{\sqrt{2}}{2}r$

Let $x = AB$, the side of the square.

Then: . $x \:=\:RO + OE \:=\:r + \tfrac{\sqrt{2}}{2}r$

Therefore: . $x \:=\:\left(\tfrac{2 + \sqrt{2}}{2}\right)r$
• Feb 2nd 2013, 09:20 PM
Jammix
Re: Circumscribe a square about a semicircle!
Can you give me the steps in which you constructed that? Thanks! :)
• Feb 3rd 2013, 08:34 AM
Soroban
Re: Circumscribe a square about a semicircle!

Correction: . $\Delta OE\color{red}{P}$ is an isosceles right triangle.

(I was not allowed to edit my post. Why?)