Question

the lenght of sides of right angled triangle are 5x cm and (3x-1)cm. if the area of triangle is 60cm*2,find its hypotenuse​

Answer 1

Answer

• The length of the Hypothenuse is 31 cm

Explanation

Given

• Length of two sides of a right angled triangle are 5x & 3x-1
• Area of the triangle is 60 cm²

To Find

• Length of the hypotenuse

Solution

Here we shall first use the Pythagoras theorem and find a equation for the Hypotenuse then using the formula to find area find the value of x and substitute this in the equation for the Hypotenuse to find the answer!!

✭ Using Pythagoras Theorem

➝ Hypotenuse² = Base² + Side²

➝ Hypotenuse² = 5x²+(3x-1)²

➝ Hypotenuse = 5x+3x-1

➝ Hypotenuse = 8x-1

✭ The Value of x

➝ Area of ∆ = ½ × Base × Height

➝ 60 = ½ × 5x × (3x-1)

➝ 60 = ½ × 15x²-5x

➝ 60 × 2 = 15x²-5x

➝ 120 = 15x²-5x

➝ 15x²-5x-120 = 0

[Dividing by 5]

➝ 3x²-x-40 = 0

➝ 3x²-12x+10x-40 = 0

➝ 3x(x-4) + 10(x-4) = 0

➝ (3x+10)(x-4)

➝ x = -10/3 or 4

[Ignoring negatively]

➝ x = 4

✭ Length of Hypotenuse

➝ 8x-1

➝ 8×4-1

➝ 32-1

➝ Hypotenuse = 31 cm

Answer 2

Given

• The length of two sides are 5x cm and (3x - 1) cm.
• Area of triangle is 60 cm².

To find

• Length of the hypotenuse.

Figure

$\setlength{\unitlength}{1cm}\begin{picture}(6,5)\linethickness{.4mm}\put(1,1){\line(1,0){4.5}}\put(1,1){\line(0,1){3.5}}\qbezier(1,4.5)(1,4.5)(5.5,1)\put(.3,2.5){\large\bf 3x-1}\put(2.8,.3){\large\bf 5x}\put(1.02,1.02){\framebox(0.3,0.3)}\put(.7,4.8){\large\bf A}\put(.8,.3){\large\bf B}\put(5.8,.3){\large\bf C}\qbezier(4.5,1)(4.3,1.25)(4.6,1.7)\put(3.8,1.3){\large\bf \Theta }\end{picture}$

Solution

★ Using Pythagoras theorem

$\underline{\boxed{Hypotenuse^2 = Base^2 + Side^2}}$

→ AC² = AB² + BC²

→ AC² = (3x - 1)² + (5x)²

Taking root both the side,

→ AC = 3x - 1 + 5x

→ AC = 8x - 1

Now,

$\underline{\boxed{Area\: of\: a\: triangle = \dfrac{1}{2} × base × height}}$

→ 60 = ½ × 5x × (3x -1)

→ 60 = ½ × 15x² - 5x

→ 120 = 15x² - 5x

→ 15x² - 5x - 120 = 0

or

→ 3x² - x - 40 = 0

Splitting the middle term,

→ 3x² - 12x + 10x - 40 = 0

→ 3x(x - 4) + 10(x - 4) = 0

→ (3x + 10) (x - 4) = 0

→ x = -10/3 [This is not possible because length of a side can't be negative]

→ x = 4

By putting the value,

» The length of hypotenuse is

⟹ Hypotenuse = (8x - 1)

⟹ Hypotenuse = 8(4) - 1

⟹ Hypotenuse = 32 - 1

⟹ Hypotenuse = 31 cm

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