In the given figure, the line segment joining the midpoints M and N of opposite sides AB and DC respectively of quadrilateral ABCD is perpendicular to both sides . Prove that the other sides of the quadrilateral are equal .
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Step-by-step explanation:
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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