Triangles- The Ambiguous Case Lily Yang- 2007

Solving Triangles If you are given: Side-Side-Side (SSS) or Side-Angle-Side (SAS), use the Law of Cosines. If you are given: Angle-Side-Angle (ASA) or Angle-Angle-Side (AAS), use the Law of Sines. If you are given: Hypotenuse-Leg (HL), You have a right triangle. Use SOHCAHTOA, the Pythagorean Theorem, or see if you have a Pythagorean triple. Here are some common triples: (3,4,5) (5,12,13) (7,24,25) (8,15,17) If you are given: Side-Side-Angle (SSA– in that order!!), then you have the AMBIGUOUS CASE!

Ambiguous Case- Rules # = “Magic Number” (Height of the triangle) S = Side opposite the given angle. If S is smaller than #, then there is no solution. If S is equal to #, you have a right triangle with one solution. If S is larger than #, you either have 1 OR 2 solutions. Here’s how to decide if you have 1 or 2 solutions: If S is larger than *, you have 1 solution. If S is smaller than *, you have 2 solutions. # S * Given angle Remember: The given angle is ACUTE. To find the Magic Number: # = * sin (Angle)

Example Find all solutions for the triangle described below: A=40° a=12 b=16 S (Side opposite) is larger than #, and S is smaller than *. Therefore, we have TWO solutions, like this: 40° A B C First, find your magic number. # = 16 sin (40°) = ° AB C AB C Solution 1 Solution 2

Solution 1 Sin B 1 = Sin 40 ° B 1 = Sin -1 ((16 Sin 40)/12) = 58.9 ° C 1 = 180 ° – 58.9 ° – 40 ° = 81.1 ° c 1 = c1 = 12 = 18.4 Sin c1 Sin A 40° A1A1 B1B1 C1C1 c1c1

Solution 2 B 2 is the supplement of B 1 ! B 2 = 180 ° ° = 121 ° C 2 = 180 ° – 121 ° – 40 ° = 19 ° c 2 = c2 = 12 = 6.1 Sin C2 Sin 40 ° ° A2A2 B2B2 C2C2 c2c2