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### 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
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry

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