Let a, b and c be the lengths of the triangle.

Let ‘a’ be the greatest side with length 17 cm and c be the smallest length.

Given, Perimeter of triangle = 40 cm

⇒ (sum of all sides) = 40

⇒ a + b + c = 40

⇒ 17 + b + c = 40

⇒ b + c = 23

Then semi-perimeter:

⇒ s = Perimeter/2

= 40/2 cm

= 20 cm

We know that,

Area of triangle = $(\sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)})$

Where, s is semi perimeter and a, b and c are sides of the triangle

$(\Rightarrow 60 = \sqrt {20 \times \left( {20-17} \right) \times \left( {20-b} \right) \times \left( {20-\left( {23-b} \right)} \right)})$

Squaring both sides

⇒ 3600 = 60 × (20 - b) × (b - 3)

⇒ 60 = -b² + 23b – 60

⇒ b² – 23b + 120 = 0

Solving above we get,

b = 15 or 8

So if b = 15cm then,

⇒ c = 23 – 15

= 8 cm

And if b = 8 cm then,

⇒ c = 23 – 8

= 15 cm