Ex.6.5 Q5 Triangles Solution - NCERT Maths Class 10

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Question

\(ABC\) is an isosceles triangle with \({AC = BC}\). If , \(A B^{2}=2 A C^{2}\) prove that \(ABC\) is a right triangle.

Diagram

Text Solution

Reasoning:

As we know ,In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Steps:

In \(\Delta A B C\)

\[\rm{A C} =\rm{B C}\]

\[\begin{align} {\rm{And}} \,A B^{2} &=2 A C^{2} \\ &=A C^{2}+A C^{2} \\ A B^{2} &=A C^{2}+B C^{2}[\because A C=B C] \end{align}\]

\(\Rightarrow \angle A C B=90^\circ\)

\(\begin{align}\Rightarrow \Delta A B C \end{align}\)  is a right triangle