InterviewSolution
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InterviewSolution
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In any triangle. `if(a^2-b^2)/(a^2+b^2)=("sin"(A-B))/("sin"(A+B))`, then prove that the triangle is either right angled or isosceles. |
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Answer» `(a^(2) - b^(2))/(a^(2) + b^(2)) = (sin (A - B))/(sin (A + B))` or `(4R^(2) sin^(2) A - 4R^(2) sin^(2) B)/(4R^(2) sin^(2) A + 4R^(2) sin^(2) B) = (sin (A - B))/(sin (A + B))` (Using Sine Rule) or `(sin (A + B) sin (A- B))/(sin^(2) A + sin^(2)B) = (sin(A - B))/(sin(A + B))` `rArr sin(A - B) = 0 " or " (sin (pi C))/(sin^(2) A + sin^(2) B) = (1)/(sin (pi - C))` or `A = B " or " sin^(2) C = sin^(2) A + sin^(2) B` or `A = B " or " c^(2) = a^(2) + b^(2)` [from the sine rule] Therefore, the triangle is isosceles or right angled |
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