Ex.6.3 Q10 Triangles Solution - NCERT Maths Class 10

Go back to  'Ex.6.3'

Question

\(CD\) and \(GH\) are respectively the bisectors of \(\angle ACB\) and \(\angle EGF\) such that \(D\) and \(H\) lie on sides \(AB\) and \(FE\) of \(\Delta ABC\) and \(\Delta EFG\) respectively. If \(\triangle ABC \sim \triangle FEG\), show that:

(i)\(\begin{align}\frac{CD}{GH}=\frac{AC}{FG}\end{align}\)

(ii)\(\begin{align}\text{ }\Delta DCB\text{ }\sim{\ }\text{ }\Delta HGE\end{align}\)

(iii) \(\begin{align}\Delta DCA\text{ }\sim{\ }\Delta HGF\end{align}\)

Diagram

 

Text Solution

  

(i) Reasoning:

If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

This may be referred to as the \(AA\) similarity criterion for two triangles.

Steps:

\(\angle ACB\,=\,\angle FGE\)

\( \Rightarrow \frac {{\angle ACB}}{2}= \frac {{\angle FGE}}{2}\)

\(\Rightarrow \angle ACD\,=\,\angle FGH\)    (\(CD \) and GH are bisectors of \(\angle C {\,\rm{and}} \,\angle G\) respectively)

In \(\Delta ADC\,{\rm{and}}\,\Delta FHG\)

\[\begin{align}\angle D A C&=\angle H F G\,\,[\because \Delta A D C \sim \Delta F E G]\\ \end{align}\]

\(\angle A C D=\angle F G H\) 

\[\begin{align} \Rightarrow \Delta A D C\,\sim\,\Delta F H G \end{align}\]   (\(AA\) criterion)

[If two triangles are similar, then their corresponding sides are in the same ratio]

\[\begin{align} \Rightarrow \frac{C D}{G H}=\frac{A C}{F G}\end{align}\]

(ii) Reasoning:

If two angles of one triangle are respectively equal to two angles of another triangle,then the two triangles are similar.

This is reffered as AA criterion for two triangles. 

Steps:

In \(\Delta DCB\) and \(\Delta HGE\)

\[\begin{align}\angle DBC&=\angle HEG\,\left(\because \Delta ABC\sim \Delta FEG \right) \\  \angle DCB&=\angle HGE\,\left(\because \frac{\angle ACB}{2}=\frac{\angle FGE}{2} \right) \\ \Rightarrow\qquad \Delta DCB &\sim \Delta EHG(AA\,\text{criterion}) \\ \end{align}\]

(iii) Reasoning:

If two angles of one triangle are respectively equal to two angles of another triangle,then the two triangles are similar.

This is reffered as AA criterion for two triangles. 

Steps:

In \(\Delta DCA,\,\,\Delta HGF\)

\[\begin{align}& \angle DAC=\angle HFG\,\,\left[ \because\Delta ABC\sim \Delta FEG \right] \\ & \angle ACD=\angle FGH\,\,[\because \frac{{\angle ACB}}{2} = \frac {{\angle FGE}}{2}] \\ \Rightarrow\qquad &\Delta DCA\sim \Delta HGF\,\,({AA}\,\text{criterion}) \\ \end{align}\]