The lengths of the sides of a right triangle are (2x − 1) m, (4x) m and (4x + 1) m, where x > 0. Find :

(i) the value of x,

(ii) the area of the triangle.

(i) Given,

Length of three sides of right angled triangle are (2x - 1) m, (4x) m, (4x + 1) m.

The largest side is hypotenuse = (4x + 1) m

By pythagoras theorem,

⇒ (2x - 1)² + (4x)² = (4x + 1)²

⇒ 4x² - 4x + 1 + 16x² = 16x² + 8x + 1

⇒ 4x² - 4x + 1 + 16x² - 16x² - 8x - 1 = 0

⇒ 4x² - 12x = 0

⇒ 4x(x - 3) = 0

⇒ 4x = 0 or x - 3 = 0

⇒ x = 0 or x = 3

Since, x > 0.

∴ x = 3.

Hence, value of x = 3.

(ii) Calculating the sides of triangle,

Length of first side = 2x - 1 = 2(3) - 1

= 6 - 1

= 5 m

Length of second side = 4x

= 4(3)

= 12 m

Length of hypotenuse = 4x + 1

= 4(3) + 1

= 13 m

By formula,

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

= $\dfrac{1}{2}$ × 5 × 12

= 30 m².

Hence, area of triangle = 30 m².