The hypotenuse of a right triangle is 10cm long. If one leg is 2 units longer than the other leg, then how long is the longer leg?
Answers
Given
Hypotenuse of right angled ∆ = 10 cm.
One leg of ∆ is 2 units longer than the other leg.
To find
Length of longer leg of ∆
Solution
Let the length of shorter leg be y cm & length of longer leg be (y + 2) cm.
We know that in a right angled ∆ :
⇒ Hypotenuse² = Base² + Altitude²
Putting values :
⇒ (10)² = y² + (y + 2)²
⇒ 100 = y² + y² + 4y + 4
⇒ 2y² + 4y + 4 = 100
⇒ 2(y² + 2y + 2) = 100
⇒ y² + 2y + 2 = 100/2
⇒ y² + 2y + 2 - 50 = 0
⇒ y² + 2y - 48 = 0
⇒ y² + 8y - 6y - 48 = 0
⇒ y(y + 8) - 6(y + 8) = 0
⇒ (y + 8)(y - 6) = 0
⇒ y = -8 or, y = 6 [Ignoring negative one]
⇒ y = 6
Finding legs of ∆ :
⇾ Shorter leg = y = 6 cm.
⇾ Longer leg = (y + 2) = 6 + 2 = 8 cm.
Therefore,
Length of longer leg of triangle is 8 cm.
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Given :-
◘ The hypotenuse of a right triangle is 10 cm long.
◘ One leg is 2 units longer than the other leg.
To find :-
The length of the longer leg.
Solution :-
Let the length of the shorter leg be x units.
Then the length of the longer leg becomes = 2 + x units.
The length of the hypotenuse = 10 units.
As it is a right triangle, so we apply the Pythagoras Theorem* :
(H)² = (P)² + (B)²
⇒(10)² = (x)² + (x + 2)²
⇒100 = x² + x² + 4 + 4x
⇒100 = 2x² + 4x + 4
⇒x² + 2x + 2 = 50
⇒x² + 2x - 48 = 0
⇒x² + 8x - 6x - 48 = 0
⇒x(x + 8) - 6(x + 8) = 0
⇒(x + 8)(x - 6) = 0
⇒(x + 8) = 0 or (x - 6) = 0
⇒x = -8 units or x = 6 units
Length can never be negative.
So, the length of the shorter leg is x = 6 units.
Now, length of the longer leg = (x + 2) units = (6 + 2) units
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* Pythagoras Theorem : The square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides.