Inscribing a Square within a Half-Circle

Date: 04/05/2002 at 02:21:55
From: Michael Wright
Subject: Inscribing a square within a half-circle

Hello, 

I was wondering if there was a way to inscribe a square within 
a given half-circle.  I have tried a few ways, e.g. using midpoints of 
the diameter of the circle and constructing perpendiculars, but 
nothing seems to work. I'm hoping you can help me with this matter.

Thank you,
Michael

Date: 04/05/2002 at 03:52:53
From: Doctor Floor
Subject: Re: Inscribing a square within a half-circle

Hi, Michael,

Thanks for your question.

Start with the center C of the semicircle. From there draw a radius  
perpendicular to the bounding diameter DB. Now lines from C to the two 
vertices on the arc part of the semicircle must have slopes 2 and -2 
respectively if we consider DB to be the x-axis. So we choose a point 
E on the perpendicular radius ("y-axis"), and make a square with side 
CE, and add a second congruent square at the top of the first. 

 

In the figure line CK has slope -2, and meets the arc in L. Now L is 
one of the vertices of the square. We drop an altitude LN to BD, and 
the rest is easy.

If you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/