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

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

Question

(A) For a given perimeter,an equilateral triangle has the maximum area.Let the sides be $a, b, c$. The perimeter is $2s = a + b + c$,so the semi-perim

Vedclass · Accepted Answer

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

Vedclass · Answer

(A) For a given perimeter,an equilateral triangle has the maximum area.Let the sides be $a, b, c$. The perimeter is $2s = a + b + c$,so the semi-perimeter is $s$.For an equilateral triangle,$a = b = c = \frac{2s}{3}$.The maximum area $A_{max} = \frac{\sqrt{3}}{4} \left(\frac{2s}{3}\right)^2 = \frac{\sqrt{3}}{4} \cdot \frac{4s^2}{9} = \frac{s^2}{3\sqrt{3}}$.Since any triangle with perimeter $2s$ has an area $A \le A_{max}$,we have $A \le \frac{s^2}{3\sqrt{3}}$.Alternatively,using Heron's formula: $A = \sqrt{s(s-a)(s-b)(s-c)}$.By the $AM$-$GM$ inequality,$\frac{(s-a) + (s-b) + (s-c)}{3} \ge \sqrt[3]{(s-a)(s-b)(s-c)}$.Since $(s-a) + (s-b) + (s-c) = 3s - (a+b+c) = 3s - 2s = s$,we have $\frac{s}{3} \ge \sqrt[3]{(s-a)(s-b)(s-c)}$.Cubing both sides: $\frac{s^3}{27} \ge (s-a)(s-b)(s-c)$.Multiplying by $s$: $A^2 = s(s-a)(s-b)(s-c) \le \frac{s^4}{27}$.Taking the square root: $A \le \frac{s^2}{\sqrt{27}} = \frac{s^2}{3\sqrt{3}}$.

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

Similar Questions

In a triangle $ABC$,if $\tan \left(\frac{A-B}{2}\right) = \frac{1}{3} \tan \left(\frac{A+B}{2}\right)$,then $a : b =$

In $\triangle ABC$, if $\angle C = \frac{\pi}{3}$, then the value of $\cos^2 A + \cos^2 B - \cos A \cos B$ is...

In a $\Delta ABC$,$a, c, A$ are given and $b_1, b_2$ are two values of the third side $b$ such that $b_2 = 2b_1$. Then $\sin A = $

In a $\Delta ABC$,evaluate $\frac{b \sin(C - A)}{c^2 - a^2} + \frac{c \sin(A - B)}{a^2 - b^2}$.

Let $a=BC, b=CA, c=AB$ be the side lengths of a $\triangle ABC$ and $m$ be the length of the median through $A$. If $a=8, b-c=2, m=6$,then the nearest integer to $b$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module