The line-segment joining the mid-points $M$ and $N$ of opposite sides $A B$ and $D C$ of quadrilateral $A B C D$ is perpendicular to both these sides.Prove that the other sides of the quadrilateral are equal.

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Solution

Construction Join - CM and DM
In ΔCMN and ΔDMN
MN=MN ( Common)
∠CNM = ∠DNM = 90° (MN Is perpendicular to DC)
CN = DN (Since N is the mid point of DC)
By SAS congruency Δ CMN ≅Δ DMN
Therefore,
CM = DM (CPCT)
∠CMN = ∠ DMN (CPCT)
∠AMN = ∠BMN = 90 (Since MN is perpendicular to AB)
So, ∠AMN − ∠CMN = ∠BMN − ∠ DMN (Since ∠CMN = ∠ DMN )
∠AMD = ∠ BMC
In Δ AMD and Δ BMC DM = CM (Proved above)
∠AMD = ∠ BMC (Proved above)
AM = BM (M is the mid point of AB)
By SAS congruency
Δ AMD ≅Δ BMC
Therefore, AD = BC (CPCT) Hence Proved

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The line-segment joining the mid-points $M$ and $N$ of opposite sides $A B$ and $D C$ of quadrilateral $A B C D$ is perpendicular to both these sides.Prove that the other sides of the quadrilateral are equal.