In quadrilateral ABCD, BN and DM are drawn perpendicular to AC such that BN = DM. Prove that O is the midpoint of BD

Step-by-step explanation:

Given :

In the given quadrilateral ABCD,

BN and DM are the perpendiculars drawn to AC such that,

BN = DM

To prove:

Point O is the midpoint of segment BD.

Or

OD = OB

Solution:

In ΔOMD and ΔONB,

∠MOD ≅ NOB [Vertical angles]

∠M ≅ ∠N ≅ 90° [Given]

Therefore, by AA property of similarity,

ΔOMD ~ Δ ONB

Therefore, their corresponding sides will be proportional,

Since BN = DM,

OD = OB

Hence O is the midpoint of BD.

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