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/
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