# 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 $\displaystyle ABCD$ is circumscribed about the (tilted) semicircle.
The circle has center $\displaystyle O$ and radius $\displaystyle r\!:\;OP = OQ = OR = OS = r.$

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

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

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

Therefore: .$\displaystyle x \:=\:\left(\tfrac{2 + \sqrt{2}}{2}\right)r$
• 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: .$\displaystyle \Delta OE\color{red}{P}$ is an isosceles right triangle.

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