The sides of a right triangle containing the right angle are 5x cm, (3x - 1) cm. If the area of the triangle be $60 \text{ cm}^2$, calculate the length of the sides of the triangle.

Consider △ABC as a right angled triangle.

AB = 5x cm and BC = (3x - 1) cm

We know that,

Area of △ABC = ${1}{2}$ × base × height = $\dfrac{1}{2}$ × BC × AB

Substituting the values we get,

⇒ 60 = $\dfrac{1}{2}$ × (3x - 1) × 5x

⇒ 120 = 5x(3x - 1)

⇒ 120 = 15x² - 5x

⇒ 15x² - 5x - 120 = 0

⇒ 5(3x² - x - 24) = 0

⇒ 3x² - x - 24 = 0

⇒ 3x² - 9x + 8x - 24 = 0

⇒ 3x(x - 3) + 8(x - 3) = 0

⇒ (3x + 8)(x - 3) = 0

⇒ 3x + 8 = 0 or x - 3 = 0

⇒ 3x = -8 or x = 3

⇒ x = $-\dfrac{8}{3}$ or x = 3.

Since, x cannot be negative as length of a side cannot be negative. So, x = 3.

AB = 5 × 3 = 15 cm

BC = (3 × 3 - 1) = 9 - 1 = 8 cm

In right angled △ABC,

Using Pythagoras theorem,

Hypotenuse² = Perpendicular² + Base²

⇒ AC² = AB² + BC²

Substituting the values we get,

⇒ AC² = 15² + 8²

⇒ AC² = 225 + 64

⇒ AC² = 289

⇒ AC = $\sqrt{289}$

⇒ AC = 17 cm.

Hence, the length of the sides of the triangle is 17 cm, 15 cm and 8 cm.