The lengths of the sides of a triangle are al | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the sides be $a = \alpha - \beta$,$b = \alpha + \beta$,and $c = \sqrt{3\alpha^2 + \beta^2}$.Since $\alpha > \beta > 0$,we observe that $c^2 =

Vedclass · Accepted Answer

$\frac{2\pi}{3}$

Vedclass · Answer

(C) Let the sides be $a = \alpha - \beta$,$b = \alpha + \beta$,and $c = \sqrt{3\alpha^2 + \beta^2}$.Since $\alpha > \beta > 0$,we observe that $c^2 = 3\alpha^2 + \beta^2$ is the largest square,as $a^2 + b^2 = 2\alpha^2 + 2\beta^2  \beta^2$).Thus,$c$ is the longest side,and the angle $C$ opposite to $c$ is the largest angle.Using the Law of Cosines: $\cos C = \frac{a^2 + b^2 - c^2}{2ab}$.Substituting the values: $\cos C = \frac{(\alpha - \beta)^2 + (\alpha + \beta)^2 - (3\alpha^2 + \beta^2)}{2(\alpha - \beta)(\alpha + \beta)}$.$\cos C = \frac{(\alpha^2 - 2\alpha\beta + \beta^2) + (\alpha^2 + 2\alpha\beta + \beta^2) - 3\alpha^2 - \beta^2}{2(\alpha^2 - \beta^2)}$.$\cos C = \frac{2\alpha^2 + 2\beta^2 - 3\alpha^2 - \beta^2}{2(\alpha^2 - \beta^2)} = \frac{\beta^2 - \alpha^2}{2(\alpha^2 - \beta^2)}$.$\cos C = \frac{-(\alpha^2 - \beta^2)}{2(\alpha^2 - \beta^2)} = -\frac{1}{2}$.Since $\cos C = -\frac{1}{2}$,the angle $C = \frac{2\pi}{3}$.

The lengths of the sides of a triangle are $\alpha - \beta$,$\alpha + \beta$,and $\sqrt{3\alpha^2 + \beta^2}$,where $\alpha > \beta > 0$. Its largest angle is:

Similar Questions

In a $\Delta ABC,$ if $A:B:C = 3:5:4,$ then $a + b + c\sqrt{2}$ is equal to

In a $\Delta ABC,$ if $3a = b + c,$ then the value of $\cot \frac{B}{2} \cot \frac{C}{2}$ is

If $a, b, c$ are the sides and $A, B, C$ are the angles of a triangle $ABC$,then $\tan \left( \frac{A}{2} \right)$ is equal to

In $\triangle ABC, (r_2 + r_3) \cot \left(\frac{B+C}{2}\right) = $

If in $\triangle ABC$,with usual notations,$a \cdot \cos^2 \frac{C}{2} + c \cdot \cos^2 \frac{A}{2} = \frac{3b}{2}$,then

Vedclass Test Series

Exam Paper Generator

Online Exam Module