In the adjoining figure, BM and DN are perpendiculars to AC such that BM = DN then AC bisects BD.

True

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False

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Solution

The correct option is A True
ABCD is a quadrilateral in which diagonals AC and BD intersect each other at R.

Also, $D N ⊥ A C$ and $B M ⊥ A C$ . $D N = B M$ .

In $△ D N R$ and $△ B M R$ , we have,

$∠ D R N = ∠ B R M$ [ Vertically opposite angles ]

$∠ D N R = ∠ B M R$ [ 90 each ]

$D N = B M$ [ Given ]

$△ D N R ≅ △ B M R$ (AAS)

$D R = B R$ [ By CPCT ]

$⟹$ AC bisects BD.

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In Fig. BM and DN are both perpendiculars to the segments AC and BM = DN. Prove that AC bisects BD [3 MARKS]

In the adjoining figure, $A B C D$ is a parallelogram.
$B M ⊥ A C a n d D N ⊥ A C$ .
Prove that:
$Δ B M C ≅ Δ D N A$ and,
$B M = D N$

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