Ex.6.4 Q4 Triangles Solution - NCERT Maths Class 10

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If the areas of two similar triangles are equal, prove that they are congruent.


 Video Solution
Ex 6.4 | Question 4

Text Solution


As we know that two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio.

The ratio of the areas of two similar trianglesis equal to the square of the ratio of their corresponding sides.   

As we know if three sides oh one triangle are equal to the three sides of another triangle,then the two triangles are congruent.


\(\Delta ABC \sim \Delta DEF\)

\[\begin{align}\Rightarrow \quad&\frac{{AB}}{{DE}} = \frac{{BC}}{{EF}} = \frac{{CA}}{{FD}}\\ & \text{(SSS Criterion)} \end{align}\]

\[\begin{align}&\;ar\Delta ABC=\Delta DEF\\\Rightarrow\quad&\frac{{ar\Delta ABC}}{{ ar\Delta DEF}} = 1\;\;\dots(1)\end{align}\]


\[\begin{align}\frac{{ar\Delta ABC}}{{ar\Delta DEF}} &= \frac{{{{(AB)}^2}}}{{{{(DE)}^2}}} \\= \frac{{{{(BC)}^2}}}{{{{(EF)}^2}}} &= \frac{{{{(CA)}^2}}}{{{{(FD)}^2}}}\end{align}\]

From \((1)\)

\[\begin{align}{\frac{(A B)^{2}}{(D E)^{2}}}&=\frac{(B C)^2}{(E F)^2} \\ & =\frac{(C A)^2}{(F D)^2} \\ & =1 \\ {\Rightarrow \; \frac{(A B)^{2}}{(D E)^{2}}}&={1} \\ {\Rightarrow \;\; (AB)^{2}}&={(D E)^{2} }\\ \Rightarrow \;\;\quad A B &=DE \;\;\ldots (2)\end{align}\]


\( \Rightarrow\)   \( BC = EF .... (3)\)

\( \Rightarrow\)   \(CA =FD …. (4)\)

Now, in \(\Delta ABC,\,\Delta DEF\)

\( \Rightarrow\)\(AB = DE\)    \(\text{(from 2)}\)

\( \Rightarrow\)\(BC = EF\)    \(\text{(from 3)}\)

\( \Rightarrow\)\(CA = FD\)    \(\text{(from 4)}\)

\[\begin{align} \Rightarrow\quad &\Delta ABC \cong \Delta DEF\\& \text{(SSS congruency)}\end{align}\]

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