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
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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
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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,
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:
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
⇢ 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
⇢ (Base + Height)² = Base² + Height² + 2 × Base × Height
⇢ (17)² = Hypotenuse² + 2 × 60
⇢ 289 = Hypotenuse² + 120
⇢ 289 - 120 = Hypotenuse²
⇢ 169 = Hypotenuse²
⇢ Hypotenuse = 13 cm
∴ Hence, Hypotenuse of triangle is 13 cm.