How do you inscribe a square into a given semi-circle?

and i need the proof also.

please and thanks.

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- Feb 15th 2009, 03:16 PMPabo258inscribe square into semicircle?
How do you inscribe a square into a given semi-circle?

and i need the proof also.

please and thanks. - Feb 15th 2009, 03:43 PMNacho
Inscribe a rectangle of wide X and long 2X in a circunference...

- Feb 15th 2009, 05:32 PMSoroban
Hello, Pabo258!

Quote:

How do you inscribe a square into a given semicircle?

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 . . .

- Jan 16th 2010, 05:44 AMGarasProof
Does anyone know how to prove this?

- Jan 16th 2010, 04:32 PMArchie Meade
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. - Jan 16th 2010, 04:46 PMArchie Meade
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 AMWilmer
Area of inscribed square in semicircle radius r = x

Area of inscribed square in circle radius r = y

x/y = 2/5