JEE Advanced2020MathematicsProperties of TrianglesActual
Let x , y and z be positive real numbers. Suppose x , y and z are the lengths of the sides of a triangle opposite to its angles X , Y and Z respectively. If tan X 2 + tan Z 2 = 2 y x + y + z , then which of the following statements is/are TRUE?
Options
- A2 Y = X + Z
- BY = X + Z
- Ctan X 2 = x y + z
- Dx 2 + z 2 − y 2 = x z
Correct answer
B. Y = X + Z
Step-by-step solution
We have, tan X 2 + tan Z 2 = 2 y x + y + z   &   s = x + y + z = perimeter of a triangle Apply half-angle formula and x + y + z = s then, ⇒ ∆ s s − x + ∆ s s − z = y s ⇒ ∆ = s − x s − z ⇒ ∆ 2 = s s − x s − y s − z = s − x 2 s − z 2 ⇒ y 2 = x 2 + z 2 ⇒ ∠ Y = 90 ° ⇒ ∠ Y = ∠ X + ∠ Z An option B is correct. Now, x y + z = sin X sin Y + sin Z = sin X 1 + sin π