The sides of a triangle are in the ratio $\dfrac{1}{2} : \dfrac{1}{3} : \dfrac{1}{4}$ and its perimeter is 91 cm. Find the lengths of the sides of the triangle.

Given,

Side₁ : Side₂ : Side₃ = $\dfrac{1}{2} : \dfrac{1}{3} : \dfrac{1}{4}$

= $\dfrac{1}{2} \times 12 : \dfrac{1}{3} \times 12 : \dfrac{1}{4} \times 12$

= 6 : 4 : 3.

Let Side₁ = 6a and Side₂ = 4a and Side₃ = 3a

Sum of sides of triangle = 6a + 4a + 3a = 13a

Given,

Perimeter of triangle = 91 cm

⇒ 13a = 91

⇒ a = $\dfrac{91}{13}$

⇒ a = 7.

Therefore,

Length of Side₁ = 6a = 6(7) = 42 cm

Length of Side₂ = 4a = 4(7) = 28 cm

Length of Side₃ = 3a = 3(7) = 21 cm

Hence, the lengths of the sides are 42 cm, 28 cm and 21 cm.