Q5 of 109 Page 287

In the adjoining figure, BM⊥AC and DN⊥AC. If BM = DN, prove that AC bisects BD.

Given: In ABCD, in which BM ​⊥ AC and DN ⊥ AC and BM = DN.


To prove: bisects BD ie. DO = BO


Proof:


Now, in ∆OND and ∆OMB, we have,
∠OND = ∠OMB …90° each
∠DON = ∠ BOM …Vertically opposite angles
Also, DN = BM …Given
Hence, by AAS congruence rule,


∆OND ≅ ∆OMB
∴ OD = OB …CPCT
​Hence, AC bisects BD.


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