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1. 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 ?

A. 3 cm

B. 4 cm

C. 5 cm

D. 6 cm

Solution:

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{aligned} Area = \sqrt{s (s - a) (s - b) (s - c)} \\ = \sqrt{15 \times 2 \times (15 - x) (x - 2)} \\ = \sqrt{30 (15 - x) (x - 2)} \\ ∴ 30 (15 - x) (x - 2) = {(30)}^{2} \\ \Rightarrow (15 - x) (x - 2) = 30 \\ \Rightarrow x^{2} - 17 x + 60 = 0 \\ \Rightarrow (x - 12) (x - 5) = 0 \\ \Rightarrow x = 12 or x = 5 \\ Smallest side  =  5 cm \end{aligned}

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