In the figure, $P A $ and $ P B $ are tangents from an external point $ P $ to a circle with centre $ O $. $ L N $ touches the circle at $ M $. Prove that $ P L+L M=P N+M N $."

Given:

In the figure, \(P A \) and \( P B \) are tangents from an external point \( P \) to a circle with centre \( O \). \( L N \) touches the circle at \( M \).

To do:

We have to prove that \( P L+L M=P N+M N \).

Solution:

$PA$ and $PB$ are tangents to the circle from $P$.

This implies,

$PA = PB$

Similarly,

$LA$ and $LM$ are tangents from $L$.

$LA = LM$

$NB = NM$

Therefore,

$PA = PB$

$\Rightarrow PL + LA = PN + NB$

$PL + LM = PN + NM$

Hence proved.

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Updated on: 10-Oct-2022

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