I was working on a geometry problem relating to the angle bisectors of triangles :In triangle Δ A B C, ∠ A = 40 ° , ∠ B = 20 ° , and A B − B C = 4 . Find the length of angle bisector from ∠ C .I was able to figure out a majority of the angle measures, but I was unable to utilize the information about the side lengths to find the angle bisector.Does anyone what method I have to use to solve this?

Question

I was working on a geometry problem relating to the angle bisectors of triangles :In triangle   Δ     A  B  C,          ∠    A  =  40      °    ,         ∠    B  =  20      °    ,     and       A  B  −  B  C  =  4   . Find the length of angle bisector from           ∠    C    .I was able to figure out a majority of the angle measures, but I was unable to utilize the information about the side lengths to find the angle bisector.Does anyone what method I have to use to solve this?

Patricia Curry · Accepted Answer

Second solution by using sine rule: Let the angle bisector intersects with |AB| on D                    sin        ⁡                  C                            sin        ⁡                  A                      =                    sin        ⁡                              120                          ∘                                                  sin        ⁡                              40                          ∘                                            =                    sin        ⁡        (                              40                          ∘                                      +                              80                          ∘                                      )                    sin        ⁡                              40                          ∘                                            =  cos  ⁡      80          ∘        +  2      cos    2    ⁡      40          ∘        =  1  +  2  cos  ⁡      80          ∘        =                    x        +        4            x        =  1  +            4      x      Thus,            4      x        =  2  cos  ⁡  80Again by using sine rule on   △      B    C    D  ,            x                        |                D        C                  |                      =                    sin        ⁡                              80                          ∘                                                  sin        ⁡                              20                          ∘                                            =                    sin        ⁡                              80                          ∘                                                  sin        ⁡                              160                          ∘                                            =            1              2        cos        ⁡                              80                          ∘                                          Thus,      |    D  C      |    =  4

Davon Irwin · Accepted Answer

Let the angle bisector intersects with       |    A  B      |   on   D and take a point,   E on       |    A  B      |   such that   ∠      E    C    B    =      80          ∘      . Then,       |    B  E      |    =      |    B  C      |  ,        |    A  E      |    =  4    ∠      E    C    A    =      40          ∘        which gives us       |    E  C      |    =  4. Also   ∠      E    C    D    =      20          ∘      ,   ∠      D    E    C    =      80          ∘        and   ∠      E    D    C    =      80          ∘        Thus,       |    C  D      |    =      |    E  C      |    =  4

I was working on a geometry problem relating to the angle bisectors of triangles : In triangle

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