# 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}

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