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

Question

In the adjoining figure, BM &#x22A5; AC and DN &#x22A5; AC. If BM = DN, prove that AC bisects BD.

Philoid · Accepted Answer

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.