1.

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

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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