The perimeter of a triangle is 30 cm and its area is 30 cm2. If the largest side measures 13 cm, then what is the length of the smallest side of the triangle ?

Answer: Option C

Let the smallest side be x cm.
Then, other sides are 13 cm and (17 - x) cm
Let a = 13, b = x and c = (17 - x)
So, s = 15
$$\begin{array}{rl}& \text{Area}=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}\\ & =\sqrt{15\times 2\times \left(15-x\right)\left(x-2\right)}\\ & =\sqrt{30\left(15-x\right)\left(x-2\right)}\\ & \therefore 30\left(15-x\right)\left(x-2\right)={\left(30\right)}^2\\ & \Rightarrow \left(15-x\right)\left(x-2\right)=30\\ & \Rightarrow x^2-17x+60=0\\ & \Rightarrow \left(x-12\right)\left(x-5\right)=0\\ & \Rightarrow x=12\text{ or }x=5\\ & \text{ Smallest side = 5 cm}\end{array}$$