Question

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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Answer 1

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

Answer 2

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