If `8R^2=a^2+b^2+c^2` then show that the triangle is right angled

Options

A. right angled
B. isosceles
C. equilateral
D. none of these

Answer

A

Explanation

We have
`8R^(2) = a^(2) + b^(2) + c^(2)`
`4 R^(2) (sin^(2) A + sin^(2) B + sin^(2)C)`
or `sin^(2) A + sin^(2)B + sin^(2)C = 2`
or `1 - cos^(2) A + 1 - cos^(2) B + sin^(2) C = 2`
or `(cos^(2) A - sin^(2) C) + cos^(2) B = 0`
or `cos(A + C) cos (A -C) + cos^(2) B = 0`
or `-cosB [cos (A -C) - cosB] = 0`
or `-cos B [cos (A -C) + cos (A +C)] = 0`
or `-2 cos A cos B cos C = 0`
or `cos A = 0 " or " cos B " or " cos C = 0`
or `A = (pi)/(2) " or " B = (pi)/(2) " or " C = (pi)/(2)`