O is intersection of diagonals of the square ABCD. If M and N are midpoints of OB and CD respectively ,then ∠ A N M = ?

Question

O is intersection of diagonals of the square ABCD. If M and N are midpoints of OB and CD respectively ,then   ∠  A  N  M  =  ?

Jaelyn Levine · Accepted Answer

Step 1Notice that   △  O  A  M  ∼  △  D  A  N,   ∠  O  A  M  =  ∠  D  A  N.So,   ∠  N  A  M  =      45    ∘  .Step 2Drop a perp from N to diagonal BD.Notice that   △  T  M  N  ≅  △  O  A  MSo,   A  M  =  M  N leading to   ∠  A  N  M  =      45    ∘  .

Ilnaus5 · Accepted Answer

Step 1    △  A  N  J   is an isosceles right triangle.The following is about the second part of OP&#039;s question.Intuitively, If I drag the point N to D and M to O  [...]  then by moving N from D to C and M from O to B with constant speed, I think the angle remain       45          ∘      .Step 2Let     A  B  =  1  ,     D  N  =  z  ∈      [    0    ,          1      2        ]     and               O      M              O      B        =            D      N              D      C         so that     z  =      1    2     corresponds to the original diagram and     z  =  0   corresponds to     M  ↦  O  ,  N  ↦  D  . Then     O  M  =            O      B              D      C          D  N  =      z          2      , and:  tan  ⁡      (    ∠    A    N    D    )    =      1    z      tan  ⁡      (    ∠    M    N    C    )    =            1      +      z              1      −      z        =  −              1      +              1        z                    1      −              1        z              =  −              tan      ⁡              (                  π          4                )            +      tan      ⁡              (        ∠        A        N        D        )                    1      −      tan      ⁡              (                  π          4                )                  tan      ⁡              (        ∠        A        N        D        )                =  tan  ⁡      (    π    −          (              π        4            +      ∠      A      N      D      )        )  It follows that     ∠  M  N  C  +      π    4    +  ∠  A  N  D  =  π  , and therefore     x  =      π    4     regardless of z.

O is intersection of diagonals of the square ABCD. If M and N are midpoints of OB and CD respectively, then angle ANM=?

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