Question

In the following figure, if LM || CB and LN || CD, prove that `("AM")/("AB")=("AN")/("AD")`

Answer 1

In ΔABC, LM || CB        ...[given]

∴ Using the basic proportionality theorem, we have

`("AM")/("MB") = ("AL")/("LC")`

⇒ `("MB")/("AM") = ("LC")/("AL")`

⇒ `("MB")/("AM") + 1=("LC")/("AL") + 1`     ...[adding 1 on both sides]

⇒ `("MB" + "AM")/("AM") = ("LC" + "AL")/("AL")`

⇒ `("AB")/("AM") = ("AC")/("AL")`

⇒ `("AM")/("AB") = ("AL")/("AC")`            ...(1)

Similarly in ΔACD, LN || CD, we have

`("AL")/("AC") = ("AN")/("AD")`             ...(2)

From (1) and  (2)

`("AM")/("AB") = ("AL")/("AC") = ("AN")/("AD")`

⇒ `("AM")/("AB") = ("AN")/("AD")`          ...(Proved)