Ex.6.3 Q9 Triangles Solution - NCERT Maths Class 10

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Question

In Figure, \(ABC\) and \(AMP\) are two right triangles, right angled at \(B\) and \(M\) respectively.

Prove that:

(i) \(\begin{align}\Delta ABC\text{ }\sim{\ }\Delta AMP\end{align}\)

(ii) \(\begin{align}\frac{CA}{PA}=\frac{BC}{MP}\end{align}\)

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:

In \(\Delta \rm{ABC}\)and\(\Delta \rm{AMP}\) 

\[\begin{align} \angle ABC&=\angle AMP={{90}^{\circ }} \\  \angle BAC&=\angle MAP\,\left( \text{Common angle} \right) \\ \Rightarrow\quad \Delta ABC\,&\sim \,\Delta AMP \\ \end{align}\]

(ii) Reasoning:

As we know that the ratio of any two corresponding sides in two equiangular triangles is always the same

Steps:

In \(\Delta \rm{ABC}\; {\rm{and}}\; \Delta \rm{AMP}\)

\[\begin{align}\frac{CA}{PA}=\frac{BC}{MP}(\because \Delta ABC\sim \Delta AMP)\end{align}\]