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

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

 

Answers (1)

Given \Delta ABC

in which AC^{2}=AB^{2}+BC^{2} ______(i)

To prove = \angle B=90^{\circ}

construct a \Delta XYZ in which 

\angle y=90^{\circ} and AB=XY and BC=YZ

Now, XZ^{2}=XY^{2}+YZ^{2} (By pythagorous theorem)

            XZ^{2}=AB^{2}+BC^{2}      (\therefore AB=XY\; and\; BC=YZ)

From eq (i)

XZ=AC _____(ii)

Now, in \Delta ABC and \Delta XYZ

AB=XY

BC=YZ

AC=XZ _____from eq (ii)

By congruency \Rightarrow \Delta ABC\cong \Delta XYZ

Therefore by CPCT

\angle B=\angle Y

\Rightarrow \angle B=90^{\circ}\; \; \; \; (\because \angle Y=90^{\circ})

Hence proved

Posted by

Safeer PP

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