Step 1: Write the ratios in the form of lengths by using any variable.

            NOTE: We know that there is a common ratio for all sides of the triangle.

                    That means that if that common ratio is denoted as x

            EXAMPLE:  Lengths a = 3x

                                              b = 4x

                                              c = 6x

Step 2: Use the Law of Cosines to find the large angle

            Step 1:  Recall the "Law of Cosines"

             c^2 > a^2 + b^2,[/math]  then the angle at C

C is obtuse.

triangle angles A,B,C and sides a,b,c

                        NOTE:ab and c are sides.

                        C is the angle opposite side c

                    The Law of Cosines (also called the Cosine Rule) says:

                       c^2 = a^2 + b^2 - 2ab cos (C)

                                \cos\left(C\right)=\frac{a^2+b^2\ -\ c^2}{2ab}

            Step 2: Substitute the given measurements in the Cosine rule

                        NOTE: We could have taken each of the ratio results in another

                      order, of course. Now, what matters is that the largest angle in the

                        triangle is opposite to the longest side

                          EXAMPLE: \cos\ C\ =\ \frac{\left(3x\right)^2\ +\ \left(4x\right)^2\ -\ \left(6x\right)^2}{2\cdot3x\cdot4x}

            Step 3: Do some calculations

                      EXAMPLE: \cos\ C\ =\frac{\ 11x^2}{-24\ x^2}\ =\ -\frac{11}{24}

                                         Cos C = - 0 . 45833

            Step 4: Send the Cos to R.H.S side.

                        NOTE: Find the Cos inverse value

                          EXAMPLE: C = \cos^{-1}\ \left(-\ 0.4833\right)

Step 3: Conclude that the triangle type based on the angle.


             c^2 = a^2 + b^2, then the angle at C

                          C is a right angle.

             c^2 < a^2 + b^2, then the angle at C

                              C is acute.

             c^2 > a^2 + b^2,  then the angle at C

                                C is obtuse.

           The cosine of an obtuse angle is always negative,

           so it follows that the triangle (having this type of angle) is obtuse.