Ex.6.3 Q12 Triangles Solution - NCERT Maths Class 10

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Sides \(AB\) and \(BC\) and median \(AD\) of a \(\triangle ABC\) are respectively proportional to sides \(PQ , QR\) and median \(PM \) of \(\Delta PQR\) (see the below Figure) .

Show that \(\Delta ABC\)~ \(\Delta PQR\).


 Video Solution
Ex 6.3 | Question 12

Text Solution


As we know if one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.

This is referred as \(SAS\) (Side–Angle–Side)  criterion for two triangles.


In \(\Delta ABC\) and \(\Delta PQR\) 

\[\begin{align} \frac{AB}{PQ}=\frac{BC}{QM}=\frac{AD}{PM} \,\,\,\text{[Given]} \\ \end{align}\]

\(\because\) AD  and PM are median of \(\Delta ABC\) and \(\Delta PQR\) respectively

\(\begin{align}&\Rightarrow \frac{BD}{QM}=\frac{{\frac{{BC}}{2}}}{{\frac{{QR}}{2}}}={\frac{BC}{QR}} \end{align}\)

Now In \(\Delta \rm{ABD}\,\,\text {and}\,\Delta \rm{PQM}\) 

\[\begin{align}&\frac{AB}{PQ}=\frac{BD}{QM}=\frac{AD}{PM} \\ \end{align}\]

\[\Rightarrow \Delta \rm{A B D} \sim \Delta \rm{P Q M}\]

Now in \(\Delta \rm{A B C}\,\text{and}\, \Delta \rm{P Q R}\)

\[\begin{align}&\frac{A B}{P Q}=\frac{B C}{Q R}\\&(\text {Given in the statement}) \\ &\angle A B C=\angle P Q R\\&[\because \Delta A B D\sim\Delta P Q M] \\ \Rightarrow\! \Delta A B C &\!\sim \! \Delta P Q R \,\,[\text{SAS} \text { Criterion }\!\!]\end{align}\]