# inscribe square into semicircle?

• Feb 15th 2009, 03:16 PM
Pabo258
inscribe square into semicircle?
How do you inscribe a square into a given semi-circle?
and i need the proof also.
• Feb 15th 2009, 03:43 PM
Nacho
Inscribe a rectangle of wide X and long 2X in a circunference...
• Feb 15th 2009, 05:32 PM
Soroban
Hello, Pabo258!

Quote:

How do you inscribe a square into a given semicircle?
Are you talking about a construction using Eucliden tools?

Here is one method . . .
Code:

```                        C * -                         /| :                         / | :                       /  | :           R  * * *  P/  | 2r           * - - - - o    | :         *  :        /:  * | :       *  :      / :  *| :           :      /  :    | :       *    :    /  :    * :       *----+----*----*----* -       B        O    Q    A       : -  r  - : -  r  - :```

We have a semicircle with center $O$ and radius $OA = OB = r.$

At $A$ erect a perpendicular to diameter $AB$
. . and measure off a distance $2r\!:\;AC = 2r$

Draw line segment $OC$, intersecting the semicircle at $P.$

From $P$ drop a perpendicular $PQ$ to diameter $AB.$

From $P$ draw a horizotal line, intersecting the semicircle at $R.$

Segments $PQ$ and $PR$ determine the inscribed square.

I'll let someone else provide the proof . . .

• Jan 16th 2010, 05:44 AM
Garas
Proof
Does anyone know how to prove this?
• Jan 16th 2010, 04:32 PM
On Soroban's clever diagram

$|AC|=2r$

$|OA|=r$

$|AC|=2|OA|$

Triangles OPQ and OCA are similar.

Triangles OCA is a magnified version of triangle OPQ.

Therefore

$|QP|=2|OQ|$

Hence $|QP|=|PR|$ and the shape is a square.
• Jan 16th 2010, 04:46 PM
The idea is to draw a line from the semicircle centre
with a slope of 2 and mark the point where the line touches the semicircle circumference.
Calculating the inverse tangent of 2 gives you the acute angle the line makes
with the x-axis.
However, simplest is to pick a horizontal value x,
and a vertical value 2x directly above x.
Locate that point and draw a line from the semicircle centre through it.
Picking x=r and 2x=2r is far neatest.
• Jan 17th 2010, 02:20 AM
Wilmer
Area of inscribed square in semicircle radius r = x
Area of inscribed square in circle radius r = y
x/y = 2/5