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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