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Question
In Fig. BM and DN are both perpendiculars to the segments AC and BM = DN. Prove that AC bisects BD [3 MARKS]
In Fig. BM and DN are both perpendiculars to the segments AC and BM = DN. Prove that AC bisects BD [3 MARKS]
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Solution
Process : 2 Marks
Proof : 1 Mark
In Δs BMR and DNR, we have
∠BMR = ∠DNR = 90∘ [Since, BM ⊥AC and DN ⊥ AC]
∠BRM = ∠DRN [Vert. opp. Angles]
and, BM = DN [Given]
So, by AAS criterion of congruence
ΔBMR ≅ ΔDNR
⇒ BR = DR [C.P.C.T]
⇒ R is the mid-point of BD.
Hence, AC bisects BD.
Process : 2 Marks
Proof : 1 Mark
In Δs BMR and DNR, we have
∠BMR = ∠DNR = 90∘ [Since, BM ⊥AC and DN ⊥ AC]
∠BRM = ∠DRN [Vert. opp. Angles]
and, BM = DN [Given]
So, by AAS criterion of congruence
ΔBMR ≅ ΔDNR
⇒ BR = DR [C.P.C.T]
⇒ R is the mid-point of BD.
Hence, AC bisects BD.
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