In the following figure, A D = A B. Also ∠ D A B = ∠ D C B = ∠ A E C = 90 ∘ and A E = 5. Find the area of quadrilateral ABCD. ( x 2 − 25 )

Question

In the following figure,   A  D  =  A  B. Also   ∠  D  A  B  =  ∠  D  C  B  =  ∠  A  E  C  =      90    ∘   and   A  E  =  5. Find the area of quadrilateral ABCD.      (          x      2        −    25    )

Kali Galloway · Accepted Answer

Step 1Join AC and hence see by cyclic quad.  ∠  A  C  E  =  ∠  A  C  B  =      45          o      Hence   E  C  =  5  ,  A  C  =  5      2  Step 2Thus,   2      x    2    =            (                        x          2                −        25            +      5      )        2    +  B      C    2  Now apply cosine rule in   Δ  A  B  C to get  cos  ⁡  45  =                    x        2            −      50      −      B              C        2                    10              2            B      C

Francisco Proctor · Accepted Answer

Step 1Take   ∠  B  D  C  =  θ, so   D  E  =  x  c  o  s  (  θ  +  45  )  ,  x  s  i  n  (  θ  +  45  )  =  5  ,  E  C  =  x      2    c  o  s  θ  −  x  c  o  s  (  θ  +  45  )  ,  D  C  =  x      2    c  o  s  θ  ,    B  C  =  x      2    s  i  n  θ  Δ  A  D  B  +  Δ  B  D  C  =  Δ  A  D  E  +  ◻  A  E  C  B  Δ  A  D  B  +  Δ  B  D  C  =                    x        2            2        +      x    2    s  i  n  θ  c  o  s  θ  =                    x        2            2        (  s  i  n  θ  +  c  o  s  θ      )    2  Step 2  Δ  A  D  E  +  ◻  A  E  C  B  =      x    2    s  i  n  θ  c  o  s  θ  −                    x        2                    2              s  i  n  θ  c  o  s  (  θ  +  45  )  +            5              2              x  c  o  s  θEquating both we get   x  =                    5                  2                            (        s        i        n        θ        +        c        o        s        θ        )            .Using this in                     x        2            2        (  s  i  n  θ  +  c  o  s  θ      )    2   we get area   =  25.

In the following figure, AD=AB. Also angle DAB= angle DCB= angle AEC=90^circ and AE=5. Find the area of quadrilateral ABCD.

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