Question

the sum of the side of a right angled triangle containing right angle is 17 cm. if its area is 30sq.cm.find the hypotenuse of the traingle​

Answer 1

Answer:-

Given:-

Sum of the sides of a right angled triangle containing right angle = 17 cm.

⟶ Base + Height = 17 cm.

Area of the triangle = 30 cm².

We know that,

Area of a right angled triangle = 1/2 * base * height

So,

⟹ 30 = 1/2 * base * height

⟹ 30 × 2 = base * height

⟹ 60 cm² = base * height

We know,

In a right angled triangle;

(Base)² + (Height)² = (Hypotenuse)²

[ Pythagoras Theorem ]

Also,

• (a + b)² - 2ab = a² + b²

Hence,

(Base + Height)² - 2 * base * height = (Hypotenuse)²

Putting the respective values we get,

⟹ (17)² - 2(60) = (Hypotenuse)²

⟹ 289 - 120 = (Hypotenuse)²

⟹ 169 = (Hypotenuse)²

⟹ (13)² = (Hypotenuse)²

⟹ 13 cm = Hypotenuse

∴ The length of hypotenuse of the given right angled triangle is 13 cm.

Answer 2

Answer:

• Sum of sides containing right angle = 17 cm
• Area of triangle = 30 cm²
• Hypotenuse = ?

So we know that Sides which is attached to Right Angle in right angle triangle is Base & Height

Base + Height = 17 cm

• Area of right angle triangle :

⇒ Area = ½ × Base × Height

⇒ 30 = ½ × Base × Height

⇒ 30 × 2 = Base × Height

⇒ Base × Height = 60

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We can solve it by 2 methods –

Method 1 :

Let's say we have to break 60 into two numbers whose sum is equal to 17

We can break 60 into :

⇒ 60 = (1 × 60) , (2 × 30) , (3 × 20) , (4 × 15) , (5 × 12) , (6 × 10)

Here we have only one pair whose sum is 17 i.e. (5 × 12)

Taking 5 as Base and 12 as height

⇢ Hypotenuse² = Base² + Height²

⇢ Hypotenuse² = (5)² + (12)²

⇢ Hypotenuse² = 25 + 144

⇢ Hypotenuse² = 169

• Squaring root both sides

⇢ Hypotenuse = 13 cm

∴ Hence, Hypotenuse of triangle is 13 cm.

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Method 2 :

• Let's take a mathematical equation :

⇢ (a + b)² = a² + b² + 2ab

• Let's say a = Base and b = Height

⇢ (Base + Height)² = Base² + Height² + 2 × Base × Height

• we know Base² + Height² = Hypotenuse²

⇢ (17)² = Hypotenuse² + 2 × 60

⇢ 289 = Hypotenuse² + 120

⇢ 289 - 120 = Hypotenuse²

⇢ 169 = Hypotenuse²

• Squaring root both sides

⇢ Hypotenuse = 13 cm

∴ Hence, Hypotenuse of triangle is 13 cm.