Thread: inscribe square into semicircle?

How do you inscribe a square into a given semi-circle?
and i need the proof also.
please and thanks.

Inscribe a rectangle of wide X and long 2X in a circunference...

Hello, Pabo258!

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 and radius

At erect a perpendicular to diameter
. . and measure off a distance

Draw line segment , intersecting the semicircle at

From drop a perpendicular to diameter

From draw a horizotal line, intersecting the semicircle at

Segments and determine the inscribed square.


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

Does anyone know how to prove this?

On Soroban's clever diagram







Triangles OPQ and OCA are similar.

Triangles OCA is a magnified version of triangle OPQ.

Therefore



Hence and the shape is a square.

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.

Area of inscribed square in semicircle radius r = x
Area of inscribed square in circle radius r = y
x/y = 2/5