Straightedge and Compass Constructions

Date: 12/14/98 at 17:31:28
From: Ross
Subject: Constructions using only a straightedge and compass

Dear Dr. Math,

I am involved in a project where I am supposed to construct a number of 
things using only a compass and straightedge. I need to know how to 
construct a 30, 60, 90 triangle. I need to also construct the three 
medians of a scalene triangle. How do you construct a regular octagon, 
and a square with the sides of a given length? I need to know how to 
construct an equilateral triangle inscribed in a circle. I need to know 
how to construct a perpendicular line to AB through a point P not on 
AB.

Date: 12/14/98 at 18:48:17
From: Doctor Schwa
Subject: Re: Constructions using only a straight edge and compass

For the 30, 60, 90 triangle:

You should look in your geometry book for how to construct a 
perpendicular; that gives you the 90. Your book should also tell you 
how to find a midpoint (or a perpendicular bisector). If you make a 
triangle with a right angle, and make one leg one unit long, and then 
draw a circle of radius 2 to make the hypotenuse 2 units long, the 
triangle will have angles of 30, 60, 90.

For the the three medians of a scalene triangle:

A median connects a vertex to the midpoint of the opposite side, so you 
can apply the midpoint (or perpendicular bisector) construction, and  
then draw the line connecting it to the opposite vertex.

A regular octagon:

This involves constructing a bunch of 135 = 90 + 45 degree angles.
Making a right isosceles triangle, and then copying angles, should be 
one good way to do this. Or, draw a circle, draw two perpendicular 
lines meeting in the middle, then bisect each of those angles.

A square with the sides of a given length:

Draw a segment that's twice as long, then perpendicularly bisect it to 
get the right angle you need. Then repeat that construction until 
you're done.

An equilateral triangle inscribed in a circle:

Take any point on the circle, copy your 60 degree angle there.

A perpendicular line to AB through a point P not on AB:

Try looking in  your book. If you don't find it, check a geometry book 
out in your local library or school library. This is a basic 
construction you should be able to find anywhere.  

I hope my answers are of some help to you!

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/