Question

if the sum of the sides of a triangle in given,then prove that the area is greatest when the triangle is equilateral​

Answer 1

answer:

sum of sides of triangle= we can say it is perimeter of triangle

Step-by-step explanation:

and area of triangle is√3/4a^2

if we solve that with given side

we can say that area of equilateral triangle is greater than sum of it's sides hope it is helpful

Answer 2

Answer:

As Carl pointed out, Heron's formula gives the area of a triangle with sides a, b, and c as A=s(s−a)(s−b)(s−c)−−−−−−−−−−−−−−−−−√, where s=12(a+b+c).

Let 2s=a+b+c>0, and suppose A2=s(s−a)(s−b)(s−c) is maximized for a, b, and c that are not all equal. Without loss of generality, assume a<b.

It's straightforward to show that s(s−a+b2)(s−a+b2)(s−c)>A2, which means that you'll get a triangle of larger area if you replace the unequal sides a and b with equal sides, keeping the same perimeter. This contradicts the assumption that a≠b at the maximum.You can use Heron's formula for the area of a triangle to help here.

A=s(s−a)(s−b)(s−c)−−−−−−−−−−−−−−−−−√

where s is the semi-perimeter

s=a+b+c2

To find the conditions for the maximum area, we want to compute derivatives w.r.t. a, b and c, but since they are not independent variables, we first substitute the semi-perimeter equation into the are equation, to get rid of c.

c=2s−a−b

∴A=s(s−a)(s−b)(a+b−s)−−−−−−−−−−−−−−−−−−−−√

Setting the derivative w.r.t. a to zero:

dAda=s(s−b)(2s−2a−b)2A=0

⇒2s−2a−b=0

(Note that b=s is not a solution since A≠0⇒b≠s.)

Similarly, setting the derivative w.r.t. b to zero yields

2s−2b−a=0

Solving simultaneously gives

a=b=2s3

and substituting back gives

c=2s3

So a=b=c and the triangle is equilateral.

Step-by-step explanation:

hope it's help you ✌️⬇️