# Thread: inscribe square into semicircle?

1. ## inscribe square into semicircle?

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

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

3. 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 $\displaystyle O$ and radius $\displaystyle OA = OB = r.$

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

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

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

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

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

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

4. ## Proof

Does anyone know how to prove this?

5. On Soroban's clever diagram

$\displaystyle |AC|=2r$

$\displaystyle |OA|=r$

$\displaystyle |AC|=2|OA|$

Triangles OPQ and OCA are similar.

Triangles OCA is a magnified version of triangle OPQ.

Therefore

$\displaystyle |QP|=2|OQ|$

Hence $\displaystyle |QP|=|PR|$ and the shape is a square.

6. 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.

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