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 ?

Solution(By Examveda Team)

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
$$\eqalign{ & {\text{Area}} = \sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \cr & \,\,\,\,\,\,\,\,\,\,\, = \sqrt {15 \times 2 \times \left( {15 - x} \right)\left( {x - 2} \right)} \cr & \,\,\,\,\,\,\,\,\,\,\, = \sqrt {30\left( {15 - x} \right)\left( {x - 2} \right)} \cr & \therefore 30\left( {15 - x} \right)\left( {x - 2} \right) = {\left( {30} \right)^2} \cr & \Rightarrow \left( {15 - x} \right)\left( {x - 2} \right) = 30 \cr & \Rightarrow {x^2} - 17x + 60 = 0 \cr & \Rightarrow \left( {x - 12} \right)\left( {x - 5} \right) = 0 \cr & \Rightarrow x = 12{\text{ or }}x = 5 \cr & {\text{ Smallest side = 5 cm}} \cr} $$