ABC is a triangle with a right angle at C, if the median of the side c is the geometric mean of sides a and b, prove that one of the acute angles of ABC is 15 o

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ABC is a triangle with a right angle at C, if the median of the side c is the geometric mean of sides a and b, prove that one of the acute angles of ABC is       15    o

enracant60 · Accepted Answer

Step 1From the Problem is given we have   R  =      a    b   and   sin  ⁡  (  α  )  =      a          2              a        b              =      1    2              a      b       from       c    2    =      a    b   we get by squaring       a    2    +      b    2    =  4  a  b dividing by   a  b  ≠  0 we get       a    b    +      b    a    =  4.Step 2Setting   t  =      a    b   we get the quadratic equation       t    2    −  4  t  +  1  =  0 and solving this we have       t          1      ,      2        =  2  ±      3   can you finish from here? Setting   2  −      3   in the formula you will get   sin  ⁡  (  α  )  =      1    2        2    −          3       this is the formula for   sin  ⁡  (      15          ∘        )

klastiesym · Accepted Answer

Step 1Let   C  A  =  b,   C  B  =  a,   A  B  =  c, CD be a median of ΔANC and E placed on line CD such that C is a midpoint of ED.Also, let   Φ be a circumcircle of   Δ  A  E  D and   A  C  ∩  Φ  =  {  A  ,  F  }.Since   C      D    2    =  C  D  ⋅  E  C  =  A  C  ⋅  C  F and from the given   C      D    2    =  A  C  ⋅  B  C, we get   C  F  =  C  B.Step 2In another hand,   ∡  A  =  ∡  E.Thus by law of sines for   Δ  E  C  F we obtain:            sin      ⁡      ∡      F              E      C        =            sin      ⁡      ∡      E              C      F      or             sin      ⁡      ∡      F              c      2        =            sin      ⁡      ∡      A        a  or             sin      ⁡      ∡      F              c      2        =            a      c        a  or   sin  ⁡  ∡  F  =      1    2    ..But   ∡  B  =      1    2    ∡  E  D  A  =      1    2    ∡  F and since   ∡  F  =      30          ∘       or   ∡  F  =      150          ∘      , we get the answer:   ∡  B  =      15          ∘       or   ∡  B  =      75          ∘      .

ABC is a triangle with a right angle at C, if the median of the side c is the geometric mean of sides a and b, prove that one of the acute angles of ABC is 15^circ

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