Spherical Geometry and TrianglesDate: 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 perspective. 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 http://mathforum.org/dr.math/ |
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