Mars '98 LanderDate: 06/18/97 at 14:32:07 From: Russ Brill Subject: Finding sides Given an arbitrary quadrilateral in which all interior angles and two opposite sides are known, how do I find the other sides? I have found a way to do it with three simultaneous equations, but that seems a bit much. (Draw a diagonal and write an equation for each of the two resulting triangles, and one relating the unknown interior angles of those triangles.) I know I'm a little too old, but it may interest you to know that this is a real problem for the Mars '98 lander flight software. Nobody around here can figure this one out. Date: 06/18/97 at 17:14:14 From: Doctor Wilkinson Subject: Re: Finding sides This is a problem where drawing diagonals doesn't seem to be the right thing to do. Here's a rough sketch of a solution. It needs some work to cover all the cases. I'm going to suppose that the quadrilateral is convex (I'm sure the non-convex case can be handled too). Let's call it ABCD, lettering counterclockwise. Suppose we know the lengths of AB and CD. Draw a line through A parallel to DC, and suppose it intersects BC at E. Suppose for now that E is between B and C. (I'm going to leave the other case to you). Then AEB is a triangle with one side known, one angle known, and one angle that we can easily figure out, so we can use the law of sines to find the length of AE. Now we have a new quadrilateral AEDC with known angles and known opposite sides, but now it's a trapezoid. Draw a line through C parallel to AD and intersecting AE at F. Now we know the angles of CFE and the side EF, so we can use the law of sines again to finish it off. -Doctor Wilkinson, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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