In right Δ A B C, ∠ C = 90 ∘ . E is on BC such that A C = B E. D is on AB such that D E ⊥ B C . Given that D E + B C = 1 and B D = 1 2 , Find ∠ B.

Question

In right   Δ  A  B  C,   ∠  C  =      90    ∘  . E is on BC such that   A  C  =  B  E. D is on AB such that   D  E  ⊥  B  C . Given that   D  E  +  B  C  =  1 and   B  D  =      1    2  , Find   ∠  B.

helsedel1v · Accepted Answer

Step 1Let   D  E  =  x,   A  C  =  B  E  =  y.                    D        E                    B        E              =                    A        C                    B        C                        x      y        =            y              1        −        x            Step 2  ⇒  x  =      x    2    +      y    2    =            1      4        sin  ⁡  B  =                    D        E                    B        D              =            1      2        ∴  ∠  B  =      30          ∘

Grace Jefferson · Accepted Answer

Step 1Let&#039;s for simplicity solve an equivalent question, for the same construction, but twice as large, for which given constrains are       |    B  D      |    =  1,       |    D  E      |    +      |    B  C      |    =  2.Step 2Then clearly,                               |                B        E                  |                =                  |                A        C                  |                                    =        cos        ⁡        β        ,                                      |                D        E                  |                                    =        sin        ⁡        β        ,                                      |                A        B                  |                                    =                              cos            ⁡            β                                sin            ⁡            β                          ,                                      |                B        C                  |                                    =                                            cos              2                        ⁡            β                                sin            ⁡            β                          ,                                      |                D        E                  |                +                  |                B        C                  |                                    =        2        =        sin        ⁡        β        +                                            cos              2                        ⁡            β                                sin            ⁡            β                          =                  1                      sin            ⁡            β                          ,                            sin        ⁡        β                            =                              1            2                          ,                            β                            =                  30          ∘                .

In right triangle ABC, angle C=90^circ. E is on BC such that AC=BE. D is on AB such that DE⊥BC. Given that DE+BC=1 and BD=12, find angle B.

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