In ABC if ^2 ~A+ ^2 ~B= ^2 C then prove that the triangle is a ri… - Vidyadip

Question

In $\triangle \mathrm{ABC}$ if $\sin ^2 \mathrm{~A}+\sin ^2 \mathrm{~B}=\sin ^2 \mathrm{C}$ then prove that the triangle is a right angled triangle.Question is modified

In $\triangle \mathrm{ABC}$ if $\sin ^2 \mathrm{~A}+\sin ^2 B=\sin ^2 \mathrm{C}$ then show that the triangle is a right anqled triangle.

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Answer

By sine rule,

$\frac{\sin \mathrm{A}}{a}=\frac{\sin \mathrm{B}}{b}=\frac{\sin \mathrm{C}}{c}=\mathrm{k}$

$\begin{aligned} & \therefore \sin A=k a, \sin B=k b, \sin C=k c \\ & \therefore \sin ^2 A+\sin ^2 B=\sin ^2 C\end{aligned}$

$\begin{aligned} & \therefore k^2 a^2+k^2 b^2=k^2 c^2 \\ & \therefore a^2+b^2=c^2\end{aligned}$

∴ ∆ABC is a rightangled triangle, rightangled at C.

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