In a , if ^2A+ ^2B= ^2C, show that the triangle is right angled.

Question

In a $\triangle\text{ABC,}$ if $\sin^2\text{A}+\sin^2\text{B}=\sin^2\text{C},$ show that the triangle is right angled.

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Answer

Let $\sin\text{A = ak},\sin\text{B = bk},\sin\text{C = ck}$
$\sin^2\text{A}+\sin^2\text{B}=\sin^2\text{C}$
$\Rightarrow{\text{k}}^2\text{a}^2+\text{k}^2\text{b}^2=\text{k}^2\text{c}^2$ [Using sine rule]
$\Rightarrow\text{a}^2+\text{b}^2=\text{c}^2$
Since the triangle satisfies the pythagoras theorem, therefore it is right angled.

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