If A is the area and 2s is the sum of 3 sides | vedclass

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

Question

(A) We have $2s = a + b + c$ and the area $A$ is given by Heron's formula: $A^2 = s(s - a)(s - b)(s - c)$.By the Arithmetic Mean-Geometric Mean $(AM \

Vedclass · Accepted Answer

$A \le \frac{s^2}{3\sqrt{3}}$

Vedclass · Answer

(A) We have $2s = a + b + c$ and the area $A$ is given by Heron's formula: $A^2 = s(s - a)(s - b)(s - c)$.By the Arithmetic Mean-Geometric Mean $(AM \ge GM)$ inequality for the positive terms $(s-a), (s-b), (s-c)$:$\frac{(s-a) + (s-b) + (s-c)}{3} \ge \sqrt[3]{(s-a)(s-b)(s-c)}$Substituting $a+b+c = 2s$:$\frac{3s - (a+b+c)}{3} \ge \sqrt[3]{\frac{A^2}{s}}$$\frac{3s - 2s}{3} \ge \sqrt[3]{\frac{A^2}{s}}$$\frac{s}{3} \ge \sqrt[3]{\frac{A^2}{s}}$Cubing both sides:$\frac{s^3}{27} \ge \frac{A^2}{s}$$A^2 \le \frac{s^4}{27}$Taking the square root:$A \le \frac{s^2}{3\sqrt{3}}$.

If $A$ is the area and $2s$ is the sum of $3$ sides of a triangle,then:

Similar Questions

In $\triangle ABC$,if the sides $a, b, c$ are in geometric progression and the largest angle exceeds the smallest angle by $60^{\circ}$,then $\cos B$ is equal to

In a triangle,if $r_1 = 2r_2 = 3r_3$,then $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ is equal to

$ABCD$ is a rhombus. The circumradii of $\Delta ABD$ and $\Delta ACD$ are $\frac{25}{2}$ and $25$ respectively. Then the area of the rhombus is .............. $sq. \, unit$.

The angles of a triangle are in $A.P.$ and the greatest angle is double the least angle. Find the sine of the third angle.

If $\alpha, \beta$ and $\gamma$ are the lengths of the altitudes of a $\triangle ABC$ with area $\Delta$,then $\frac{\Delta^2}{R^2}\left(\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}\right)$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module