Ex.6.3 Q12 Triangles Solution - NCERT Maths Class 10

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Question

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

Diagram

 

Text Solution

  

Reasoning:

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.

Steps:

In \(\Delta ABC\) and \(\Delta PQR\) 

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

\(\begin{align}\because AD\text{ and PM are median of }&\Delta ABC\text{ and }\Delta PQR \,\,\text{respectively} \\ &\Rightarrow \frac{BD}{QM}=\frac{{\frac{{BC}}{2}}}{{\frac{{QR}}{2}}}={\frac{BC}{QR}} \end{align}\)

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

\[\begin{align}\frac{A B}{P Q}&=\frac{B C}{Q R}\,\,\,(\text { given in the statement }) \\ \angle A B C&=\angle P Q R\;\,\,\,[\because \Delta A B D\sim\Delta P Q M] \\ \Rightarrow\quad \Delta A B C &\sim \Delta P Q R \,\,\,[S A S \text { criteion }]\end{align}\]