In the interior of a triangle ABC, a point P is marked in such a way that: P C = B C and the measure of the angle PAB is equal to the measure of the angle PAC which is 17 ° . calculate the measure of angle PCB, if the measure of angle B = 107 o

Question

In the interior of a triangle ABC, a point P is marked in such a way that:   P  C  =  B  C and the measure of the angle PAB is equal to the measure of the angle PAC which is   17      °   . calculate the measure of angle PCB, if the measure of angle   B  =      107    o

Savanah Hernandez · Accepted Answer

Step 1Since   △  A  C  D is an isosceles, from   ∠  D  A  C  =      34    ∘   we have  ∠  A  D  C  =  ∠  A  C  D  =      73    ∘  It&#039;s given that   ∠  A  C  B  =      39    ∘   , so  ∠  B  C  D  =  ∠  A  C  D  −  ∠  A  C  B  =      34    ∘  Here you might already know where this is going. With   ∠  B  C  D  =      34    ∘   , we obtain  ∠  C  B  D  =      180    ∘    −  ∠  B  C  D  −  ∠  B  D  C  =      73    ∘  and it quickly follows that  B  C  =  C  DNow, with   B  C  =  P  C and   P  D  =  P  C , we have an important conclusion, which is  P  C  =  C  D  =  P  DHence   △  P  C  D is an equilateral triangle, and   ∠  P  C  D  =      60    ∘   . This implies  ∠  A  C  P  =  ∠  A  C  D  −  ∠  P  C  D  =      13    ∘  and finally,  ∠  P  C  B  =  ∠  A  C  B  −  ∠  A  C  P  =  θ  =      26    ∘

arridsd9 · Accepted Answer

Step 1Applying Trigonometric form of Ceva&#039;s theorem,      sin    ⁡    ∠    P    A    C    ⋅    sin    ⁡    ∠    P    C    B    ⋅    sin    ⁡    ∠    P    B    A    =  sin  ⁡  ∠  A  C  P  ⋅  sin  ⁡  ∠  C  B  P  ⋅  sin  ⁡  ∠  B  A  Pi.e.          sin    ⁡          17      ∘        ⋅    sin    ⁡    θ    ⋅    sin    ⁡          (              17        ∘            +              θ        2            )        =  sin  ⁡  (      39    ∘    −  θ  )  ⋅  sin  ⁡      (          90      ∘        −          θ      2        )    ⋅  sin  ⁡      17    ∘           2    sin    ⁡          θ      2        ⋅    sin    ⁡          (              17        ∘            +              θ        2            )        =    sin    ⁡    (          39      ∘        −    θ    )    cos  ⁡      17    ∘    −  cos  ⁡  (      17    ∘    +  θ  )  =  cos  ⁡  (      51    ∘    +  θ  )      cos    ⁡          17      ∘        =    cos    ⁡    (    (          34      ∘        +    θ    )    −          17      ∘        )    +    cos    ⁡    (    (          34      ∘        +    θ    )    +          17      ∘        )        cos    ⁡          17      ∘        =    2    cos    ⁡    (          34      ∘        +    θ    )    cos    ⁡          17      ∘        cos  ⁡  (      34    ∘    +  θ  )  =      1    2    =  cos  ⁡      60    ∘    ∴  θ  =      26    ∘

In the interior of a triangle ABC, a point P is marked in such a way that: P C = B C

Answered question

Answer & Explanation

New Questions in Elementary geometry