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Spherical Geometry and Triangles

Date: 02/09/2004 at 23:33:15
From: Nil
Subject: (no subject)

Is it possible to have a triangle with two 90 degree angles, and the
other two legs from the connected 90 degree angles meet?  Where would
you find such a triangle?  I thought maybe if the triangle is on a
sphere, but then the lines aren't straight.

Date: 02/10/2004 at 00:28:04
From: Doctor Jeremiah
Subject: Re: (no subject)

Hi Nil,

You are correct that in the study of "spherical geometry" it is
possible to have a triangle with two 90 degree angles.  The idea of 
"straight" is relative to where you live.  If you live on a sphere
then something that looks straight to you will look curved from a 3D 

Let's look at an example of such a triangle using a sphere we are
familiar with--the Earth.  Say you are exactly on the North Pole. 
Walk straight south 345.8767 km.  Then walk straight east 345.8767 km.
Then walk straight north 345.8767 km.  You are back at the North Pole
and all three of your corners are 90 degree angles.

Here is how I found that particular value:

Starting on the North Pole, first you walk south X distance in a 
straight line and then you walk east X distance in a straight line.  
At that point you must be 1/4 of the way around the globe so that 
when you walk back north X distance in a straight line you will have 
a 90 degree angle with the straight line you walked south away from 
the North Pole.

The radius of the Earth is 6378.137 km, so the circumference is 
40075.02 km.  If you walk X km south from the North Pole you have 
gone X/40075.02 % of the way around the Earth which is 360*X/40075.02 
degrees.  The radius of the line of latitude at that point is 

  R = 6378.137*sin(360*X/40075.02) km.  

That means the circumference of that line of latitude is 2*Pi*R.  
When you walk east around that line of latitude you want X km to be
1/4 of the way around, so

  X = 2*Pi*R/4.  

Now we have two equations:

  R = 6378.137*sin(360*X/40075.02) 
  X = 2*Pi*R/4

And when we solve them (numerically) we get X=345.8767 km

That may be more math than you were interested in.  The short answer
to your question is that if we work with triangles drawn on a sphere
rather than on a plane, it is in fact possible for a triangle to have
two or even three 90 degree angles.

- Doctor Jeremiah, The Math Forum 
Associated Topics:
High School Non-Euclidean Geometry
High School Triangles and Other Polygons

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