First slide

Properties of triangles

Question

If $△ A B C, 8 R^{2} = a^{2} + b^{2} + c^{2}$ , then the triangle $A B C,$ is

Moderate

A

right angled
B

isosceles
C

equilateral
D

none of these

Solution

We have,
$\begin{array}{l} 8 R^{2} = a^{2} + b^{2} + c^{2} \\ \Rightarrow 8 R^{2} = 4 R^{2} (\sin^{2} A + \sin^{2} B + \sin^{2} C) \\ \Rightarrow \sin^{2} A + \sin^{2} B + \sin^{2} C = 2 \\ \Rightarrow 1 - \cos^{2} A + 1 - \cos^{2} B + \sin^{2} C = 2 \\ \Rightarrow (\cos^{2} A - \sin^{2} C) + \cos^{2} B = 0 \\ \Rightarrow \cos (A + C) \cos (A - C) + \cos^{2} B = 0 \\ \Rightarrow - \cos B {\cos (A - C) - \cos B} = 0 \\ \Rightarrow - \cos B {\cos (A - C) + \cos (A + C)} = 0 \\ \Rightarrow - 2 \cos A \cos B \cos C = 0 \\ \Rightarrow \cos A = 0 or, \cos B = 0 or, \cos C = 0 \\ \Rightarrow A = \frac{π}{2} or, B = \frac{π}{2} or, C = \frac{π}{2} \end{array}$
$\Rightarrow △ A B C$ is a right angled triangle

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