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…

Question

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 tanX/2+tanZ/2=2y/(x+y+z), then which of the following statements is/are TRUE?

MARKS Content Team · Accepted Answer

Correct answer: (B) Y=X+Z; (C) tanX/2=x/(y+z). We have, tanX/2+tanZ/2=2y/(x+y+z) & s=x+y+z=perimeter of a triangle Apply half-angle formula and x+y+z=s then, ⇒(∆)/(ss−x)+(∆)/(ss−z)=y/s⇒∆=s−xs−z ⇒∆^2=ss−xs−ys−z=s−x^2s−z^2 ⇒y^2=x^2+z^2⇒∠Y=90° ⇒∠Y=∠X+∠Z An option B is correct. Now, x/(y+z)=sinX/(sinY+sinZ)=sinX/(1+sinπ2-X)=2sinX2cosX2/(2cos^2X2)=tanX/2 An Option C is correct.

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 $\tan \frac{X}{2} + \tan \frac{Z}{2} = \frac{2 y}{x + y + z}$ , then which of the following statements is/are TRUE?

Solution

Relevant Questions

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 tanX2+tanZ2=2yx+y+z, then which of the following statements is/are TRUE?

Solution

Relevant Questions

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 $\tan \frac{X}{2} + \tan \frac{Z}{2} = \frac{2 y}{x + y + z}$ , then which of the following statements is/are TRUE?