# Ex.6.2 Q6 Triangles Solution - NCERT Maths Class 10

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## Question

In Figure, $$A, \;B$$ and $$C$$ are points on $$OP, \;OQ$$ and $$OR$$ respectively such that $$AB || PQ$$ and $$AC || PR.$$ Show that $$BC || QR.$$

## Text Solution

Reasoning:

As we know if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

Steps:

In $$\Delta OPQ$$

$$AB\,||\,PQ$$ given

\begin{align}\frac{OA}{AP}=\frac{OB}{BQ}\,\,\,.........\rm{(i)} \end{align}

$$[\because$$  Thales Theorem (BPT)]

In $$\Delta OPR$$

$$AC||PQ$$ (given)

\begin{align}\frac{OA}{AP}=\frac{OC}{CR}..............\rm{(ii)} \end{align}

[ $$\because$$Thales Theorem ($$BPT$$)]

From $$\rm (i)$$$$\rm (ii)$$

\begin{align}\frac{OA}{AP}=\frac{OB}{BR}=\frac{OC}{CR}\end{align}

\begin{align}\frac{OB}{BQ}=\frac{OC}{CR}\end{align}

Now In $$\Delta OQR$$

\begin{align} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{OB}{BQ}=\frac{OC}{CR} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,BC||QR\,\, [\because \text{ Converse of BPT ]} \\ \end{align}

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