Question

The perimeter of a triangle is $10$ cm. If one of the sides is $4$ cm.

Then the lengths of remaining sides when the area of the triangle is maximum are?

• A. $4$ cm, $2$ cm
• B. $3$ cm, $3$ cm
• C. $1$ cm, $5$ cm
• D. $4$ cm, $3$ cm

Answer 1

The correct option is B $3$ cm, $3$ cm

The area of a triangle is given by

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

where $s$ is the semi-perimeter, $a,b,c$ are the sides of the triangle.

Here,

$s=5,a=4$

Thus,

$A=\sqrt{5(5-4)(5-b)(5-c)}$

$A^2=5(5-b)(5-c)$

We also have the condition $b+c=6$ and so

$A^2=5(5-b)(5-6+b)$

$A^2=5(5-b)(b-1)$

Differentiating both sides w.r.t $b$, we get $0=5-b+(b-1)(-1)$

$\Rightarrow 0=5-b+1-b$

Or

$b=3$ and therefore, $c=3$