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

1612

First, find your magic number.

# = 16 sin (40°) = 10.3

10.3

1216

40°A B

C

40°

1612

A B

C

Solution 1 Solution 2

Solution 1

Sin B1 = Sin 40° 16 12

B1 = Sin-1 ((16 Sin 40)/12) = 58.9°

C1 = 180° – 58.9° – 40° = 81.1°

c1 = c1 = 12 = 18.4

Sin c1 Sin A

40°

1612

A1 B1

C1

c1

Solution 2

B2 is the supplement of B1!

B2 = 180° - 58.9° = 121°

C2 = 180° – 121° – 40° = 19°

c2 = c2 = 12 = 6.1

Sin C2 Sin 40°

1216

40°

A2 B2

C2

c2