Finding a triangle angle based on side length equalityConsider the triangle ABC with angle A being 70 degrees, and the side lengths satisfying: B C 2 = A C ( A B + A C )Is there any intuitive way of finding the measure of angle B?

Question

Finding a triangle angle based on side length equalityConsider the triangle ABC with angle A being 70 degrees, and the side lengths satisfying:  B      C    2    =  A  C  (  A  B  +  A  C  )Is there any intuitive way of finding the measure of angle B?

Audrey Rosales · Accepted Answer

Step 1Let   a  =  B  C  ,  b  =  A  C, and   c  =  A  B. Using the cosine rule we have:      a    2    =      b    2    +      c    2    −  2  b  c  cos  ⁡  70      º  and since       a    2    =  c  ⋅  b  +      b    2  , then:  c  ⋅  b  +      b    2    =      b    2    +      c    2    −  2  b  c  cos  ⁡  70      º    b  c  =      c    2    −  2  b  c  cos  ⁡  70      º  and this information allows you to find b in terms of c.Step 2Then substitute back into the cosine rule which allows you to relate a with b. A straightforward application of the sine rule will allow you to find angle B.I get that   B  =  35      º  .

Ashlynn Hale · Accepted Answer

Step 1From the given   B      C    2    =  A  C  (  A  B  +  A  C  ), we have from the sine rule       sin    2    ⁡  A  =  sin  ⁡  B  (  sin  ⁡  C  +  sin  ⁡  B  ).Step 2Note                                     sin          2                ⁡        A        −        sin        ⁡        B        (        sin        ⁡        C        +        sin        ⁡        B        )                            =                              sin          2                ⁡        A        −                  sin          2                ⁡        B        −        sin        ⁡        B        sin        ⁡        C                            =                              1          2                (        cos        ⁡        2        B        −        cos        ⁡        2        A        )        −        sin        ⁡        B        sin        ⁡        (        A        +        B        )                            =                    sin        ⁡        (        A        −        B        )        sin        ⁡        (        A        +        B        )        −        sin        ⁡        B        sin        ⁡        (        A        +        B        )                            =                    sin        ⁡        (        A        +        B        )        [        sin        ⁡        (        A        −        B        )        −        sin        ⁡        B        ]        =        0            which leads to   sin  ⁡  (  A  −  B  )  =  sin  ⁡  B or   B  =      1    2    A  =      35    ∘

Consider the triangle ABC with angle A being 70 degrees, and the side lengths satisfying: BC^2=AC(AB+AC).

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