Why SSA Does Not
Determine A Triangle

By Daniel Cranston

Initialization

<<Graphics`Animation`

This notebook provides a graphical animation of the three ambiguous cases of the Law of Sines.

Case 1: No solution

The first case is the case where a short side, smaller than the altitude of the triangle to be constructed leads to an impossible situation.  That is, no triangle can possibly be constructed.  The small side is the first given side, its adjacent side is the other, and the angle is on the bottom left.  The lowest side of the triangle is not given and could possibly vary.

ShowAnimation[Table[
	Graphics[
	{RGBColor[0,0,1],
	Line[{{5Cos[t], 5Sin[t]},
	{0,0}, {-15, -5Sqrt[3]}, {0, -5Sqrt[3]}}]},
PlotRange -> {{-15, 5}, {-10,0}}],
	{t, 5Pi/4, 7Pi/4, Pi/16}]];

Case 2: One solution

This is the case where the first side is equal to the altitude of the triangle.  This results in a 90‚ triangle.  Again, notice that the lower side is not one of the given fixed lengths.

ShowAnimation[Table[
	Graphics[
	{RGBColor[Floor[Abs[Sin[t]]],0,1-Floor[Abs[Sin[t]]]],
	Line[{{5Sqrt[3]Cos[t], 5Sqrt[3]Sin[t]},
	{0,0}, {-15, -5Sqrt[3]}, {0, -5Sqrt[3]}}]},
PlotRange -> {{-15, 5}, {-10,0}}],
	{t, 5Pi/4, 7Pi/4, Pi/16}]];

Case 3: Two solutions.

This is the case where the first side is longer than the altitude but shorter than the second side which is adjacent to it.  Notice that the rotating side hits the base of the triangle in two possible places to form a triangle with the given SSA.

ShowAnimation[Table[
	Graphics[{RGBColor[1-Round[1-Abs[Cos[t]]],
	0,Round[1-Abs[Cos[t]]]], 
	Line[{{10Cos[t], 10Sin[t]}, {0,0}, {-15, -5Sqrt[3]}, 
	{-5Sqrt[3]Cot[t], -5Sqrt[3]}}]},
PlotRange -> {{-15, 5}, {-10,0}}],
	{t, 4Pi/3, 5Pi/3, Pi/24}]];