Mars '98 Lander

Date: 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
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