The sides of a right-angled triangle containingthe right angle are 4x

1.	The sides of a right-angled triangle containingthe right angle are 4x cm and (2x - 1) cm. Ifthe area of the triangle is 30 cm2; calculate thelengths of its sides.
Answer» $\large{\underline{\bf{\green{Given:-}}}}$ ✦ sides of triangle = 4x , (2x -1) ✦ Area of triangle = 30cm² $\large{\underline{\bf{\green{To\:Find:-}}}}$ ✦length of each side. $\huge{\underline{\bf{\purple{Solution:-}}}}$ Area of triangle = 1/2 ×base × height ➝ 30 = 1/2 ×4x × (2x -1) ➝ 30 × 2 = 4x × (2x -1) ➝ 60 = (4x)(2x-1) ➝ 60/4 = x(2x-1) ➝ 15 = 2x² - x ➝ 2x² - x -15 = 0 Now solving EQUATION for x ➝ 2x² - 6x + 5x -15 ➝ 2x (x - 3) +5(x - 3) ➝ (2x + 5) (x -3) x = -5/2 x = 3 Length of any side cannot be negative . So, *x = 3* Base = 4x ⠀⠀⠀⠀ = 12cm Height = 2x-1 ⠀⠀⠀⠀⠀ = 5 cm we know that , according to question the triangle is RIGHT angled triangle. So, hypotenuse² = base² + perpendicular² ➝ H² = 12² +5² ➝ H² = 144 +25 ➝ H² = 169 ➝ H = √169 ➝ H = 13, So, All sides of triangle are 12 cm , 5cm ,*and* 13cm ━━━━━━━━━━━━━━━━━━━━━━━━━

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

Answer»

$\large{\underline{\bf{\green{Given:-}}}}$

$\large{\underline{\bf{\green{To\:Find:-}}}}$

$\huge{\underline{\bf{\purple{Solution:-}}}}$

Area of triangle = 1/2 ×base × height

➝ 30 = 1/2 ×4x × (2x -1)

➝ 30 × 2 = 4x × (2x -1)

➝ 60 = (4x)(2x-1)

➝ 60/4 = x(2x-1)

➝ 15 = 2x² - x

➝ 2x² - x -15 = 0

Now solving EQUATION for x

➝ 2x² - 6x + 5x -15

➝ 2x (x - 3) +5(x - 3)

➝ (2x + 5) (x -3)

Length of any side cannot be negative .

So, x = 3

Base = 4x

⠀⠀⠀⠀ = 12cm

Height = 2x-1

⠀⠀⠀⠀⠀ = 5 cm

we know that , according to question the triangle is RIGHT angled triangle.

So,

hypotenuse² = base² + perpendicular²

➝ H² = 12² +5²

➝ H² = 144 +25

➝ H² = 169

➝ H = √169

➝ H = 13,

So,

All sides of triangle are 12 cm , 5cm ,and 13cm

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