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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