The sides of a right-angled triangle containing the right angle are 4x cm and (2x - 1) cm. If the area of the triangle is 30 cm²; calculate the length of its sides.

The side not containing the right angle is hypotenuse.

Area of right-angled triangle = $\dfrac{1}{2} \times$ base × height

$= \dfrac{1}{2} \times 4x \times (2x - 1) \\[1em] = 2x(2x - 1) \\[1em] = 4x^2 - 2x$

Given, area = 30 cm²

∴ 4x² - 2x = 30

⇒ 4x² - 2x - 30 = 0

⇒ 4x² - 12x + 10x - 30 = 0

⇒ 4x(x - 3) + 10(x - 3) = 0

⇒ (4x + 10)(x - 3) = 0

⇒ 4x + 10 = 0 or x - 3 = 0

⇒ x = $-\dfrac{10}{4}$ or x = 3.

Since, side cannot be negative

∴ x ≠ $-\dfrac{10}{4}$

∴ 4x = 4(3) = 12 cm and (2x - 1) = (2(3) - 1) = 5 cm.

We know that,

⇒ (Hypotenuse)² = (Perpendicular)² + (Base)²

⇒ (Hypotenuse)² = (12)² + (5)²

⇒ (Hypotenuse)² = 144 + 25

⇒ (Hypotenuse)² = 169

⇒ Hypotenuse = $\sqrt{169}$ = 13 cm.

Hence, length of the sides of triangle are 5 cm, 12 cm and 13 cm.