Ex.6.2 Q3 Triangles Solution - NCERT Maths Class 10

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Question

In Figure if \( LM || CB\) and \(LN || CD\), prove that

\[\begin{align}\frac{AM}{AB}=\frac{AN}{AD}\end{align}\]

Text Solution

  

Reasoning:

As we know if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

Steps:

In \(\begin{align} \Delta ABC\end{align}\)

\[\begin{align} &LM||CB \\  &\frac{AM}{MB}=\frac{AL}{LC}...........\text{ }\left( \text{Eq }1 \right) \\ \end{align}\]

In \(\Delta ACD\) 

\[\begin{align}&LN||CD \\ &\frac{AN}{DN}=\frac{AL}{LC}............\left( \text{Eq }2 \right) \\ \end{align}\]

From equations \((1)\) and \((2)\)

\[\begin{align}\frac{AM}{MB}=\frac{AN}{DN}\end{align}\]

\[\begin{align}\Rightarrow \frac{MB}{AM}=\frac{DN}{AN}\end{align}\]

Adding \(1\) on both sides

\[\begin{align} \frac{MB}{AM}+1&=\frac{DN}{AN}+1 \\ \frac{MB+AM}{AM}&=\frac{DN+AN}{AN} \\ \frac{AB}{AM}&=\frac{AD}{AN} \\ \frac{AM}{AB}&=\frac{AN}{AD} \\ \end{align}\]

  
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