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Trisecting an Angle

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Date: 4/16/96 at 18:30:1
From: David Owens
Subject: help me please!

Dear Dr. Math,

My teacher gave me an assignment asking me to trisect a 90 degree
angle using only a compass and a straightedge.  He said that we
could use outside sources.  I have looked everywhere and I can't
find an answer.  I can bisect it easily but I can't trisect it
perfectly.  Would you please send me instructions on how I would
do this?

Thanks!
Aaron Isaacson
aarona@creighton.knightnet.edu
```

```
Date: 4/22/96 at 20:35:18
From: Doctor Joshua
Subject: Re: help me please!

Hi Aaron,

Okay, I think I've figured this out, but it might be hard to
explain with limited computer drawings, so bear with me, and get
paper and a pencil:

First draw your 90 degree angle, making sure that the lengths of
the two segments extending from the center are the same length
(this can be done with the compass). Call the 90 angle point P,
end of the North-South segment point Q, and the East-West segment
point R:

Q
| <-- ray of length 1
|
|
|________  <-- ray of length 1
P          R

Then draw three arcs whose radius is the same length as the
segments you've drawn, and whose centers are located at point P
(call this arc A), point Q (call this arc B), and point R (call
this arc C).  Arcs B and C should go through point P (I'm not
going to try drawing this).  Call point M the intersection of Arcs
A and B, and call point N the intersection of Arcs A and C.

Now with a straight edge, draw four straight lines:  from M to P,
M to Q, N to P, and N to R. Because the lengths of MQ=PQ=PM, and
NP=PR=RN, you have two equilateral triangles (NPR and MPQ).
Because every angle in an equilateral triangle is 60 degrees,
angle MPQ is 60 degrees, so angle MPR must be 30 (=90-60).
Similarly, angle NPQ is 30, so the last remaining angle, NPM, must
also be 30 (=90-30-30), and our right angle is trisected.

Good luck!

-Doctor Joshua,  The Math Forum

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Associated Topics:
High School Constructions
High School Euclidean/Plane Geometry
High School Geometry
Middle School Geometry
Middle School Two-Dimensional Geometry

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