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

traffig75 2022-09-23 Answered
O is intersection of diagonals of the square ABCD. If M and N are midpoints of OB and CD respectively ,then A N M = ?
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Answers (2)

Jaelyn Levine
Answered 2022-09-24 Author has 9 answers
Step 1

Notice that O A M D A N, O A M = D A N.
So, N A M = 45 .
Step 2
Drop a perp from N to diagonal BD.
Notice that T M N O A M
So, A M = M N leading to A N M = 45 .
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Ilnaus5
Answered 2022-09-25 Author has 2 answers
Step 1
A N J is an isosceles right triangle.

The following is about the second part of OP'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 2
Let 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.
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