# 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}

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