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M is a point on the side BC of a parallelogram ABCD. DM when produced meets AB produced at N. Prove that(i) DM/MN = DC/BN (ii) DN/DM = AN/DC |
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Answer» Given: ABCD is a parallelogram To prove: (i) DM / MN = DC/BN (ii) DN/DM =AN/ DC Proof: In ∆ DMC and ∆ NMB ∠DMC = ∠NMB (Vertically opposite angle) ∠DCM = ∠NBM (Alternate angles) By AAA- Similarity ∆DMC ~ ∆NMB ∴DM /MN = DC/ BN NOW, MN/DM = BN/DC Adding 1 to both sides, we get MN/DM + 1 = BN/DC + 1 ⇒ MN+DM/DM = BN+DC/DC ⇒ MN+DM/DM = BN+ AB/DC [∵ ABCD is a parallelogram] ⇒ DN/DM = AN/DC |
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