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