Ex.6.3 Q13 Triangles Solution - NCERT Maths Class 10

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Question

\(D\) is a point on the side \(BC\) of a \(\triangle ABC\) such that \(\angle ADC\) \(=\) \(\angle BAC\). Show that \(\,C{{A}^{2}}=CB.CD\) .

Diagram

 

Text Solution

  

Reasoning:

As we know if two triangles are similar then their corresponding sides are proportional.

Steps:

In \(\Delta ABC\) and \(\Delta DAC\)

\(\angle \rm{B A C}=\angle \rm{A D C}\)   (Given in the statement) 

\(\angle \rm{A C B}=\angle\rm{ A C D}\)  (Common angles)

\(\Rightarrow \Delta \rm{A B C} \sim \Delta \rm{D A C}  \,\,\,(AA criterion)\)

If two triangles are similar,then their corresponding sides are proportional

\[\Rightarrow \frac{C A}{C D}=\frac{C B}{C A}\]

\[\begin{align} {\Rightarrow C A^{2}=C B \cdot C D}\end{align}\]