JEE Advanced · Mathematics · 17. Properties of Triangles

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?

A $2 Y = X + Z$
B $Y = X + Z$
C $\tan \frac{X}{2} = \frac{x}{y + z}$
D $x^{2} + z^{2} - y^{2} = x z$

Verified Solution

Answer & Solution

Correct Answer

(C) $\tan \frac{X}{2} = \frac{x}{y + z}$

Step-by-step Solution

Detailed explanation

We have,

\tan \frac{X}{2} + \tan \frac{Z}{2} = \frac{2 y}{x + y + z} & s = x + y + z =

perimeter of a triangle Apply half-angle formula and

x + y + z = s

then,

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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	List- $I$		List- $II$
$(I)$	Probability of $(X_{2} \geq Y_{2})$ is	$(P)$	$\frac{3}{8}$
$(II)$	Probability of $(X_{2} > Y_{2})$ is	$(Q)$	$\frac{11}{16}$
$(III)$	Probability of $(X_{3} = Y_{3})$ is	$(R)$	$\frac{5}{16}$
$(IV)$	Probability of $(X_{3} > Y_{3})$ is	$(S)$	$\frac{355}{864}$
		$(T)$	$\frac{77}{432}$

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?

Step-by-step Solution

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?