Question

Of the given perimeter, the triangle having maximum area is

• A. Isosceles triangle
• B. Right angle triangle
• C. Equilateral
• D. None of these

Answer 1

The correct option is C

Equilateral

Explanation for the correct option:

Area of triangle:

Let ‘$s$’ be the constant perimeter of the triangle then, the general formula for the area of the triangle is

$A=\sqrt{s(s-a)(s-b)(s-c)}$ where, $a,b,c$ are three sides of the triangle, to get the maximum area we need to find the derivative of the area and equate it with zero to see the conditions of side relations so,

$A=\sqrt{s(s-a)(s-b)(s-c)}$ and$s-c=a+b-s$

Differentiating area $A^2$ with respect to side a keeping other side constants we get

$A^2=s(s-a)(s-b)(a+b-s)$

$\Rightarrow$$\frac{dA}{da}=\frac{s(s-b)(2s-2a-b)}{2A}$

Equate the above equation with zero we get,

$s(s-b)(2s-2a-b)=0$

Since $s=b$ can't be a solution because it leads to area zero hence,

$2s-2a-b=0$ …..(i)

Similarly differentiating again area $A$ with respect to side $b$ keeping other side constants we get,

$2s-2b-a=0$ ……(ii)

Solving both equations (i) and (ii) together we get the values,

$a=b=\frac{2s}{3}$ and using these values and relation $s=\frac{a+b+c}{2}$, we get $c=\frac{2s}{3}$

hence, all sides are equal which makes the triangle an equilateral triangle

Hence, the correct answer is option (C)