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Question
In a right-angled triangle, altitude is 2 cm longer than its base. Find the dimensions of the right-angled triangle given that the length of its hypotenuse is 10 cm.
Solution
Let x cm be the base of the right-angled triangle.
As a result, the altitudeude = (x + 2) cm.
Also, hypotenuse = 10 cm
We have the Pythagorean theorem.
(Base)2 + (Altitude)2 = (Hypotenuse)2
⇒ x2 + (x + 2)2 = 102
⇒ x2 + x2 + 4x + 4 = 100
⇒ 2x2 + 4x – 96 = 0
⇒ 2x + 2x – 48 = 0
⇒ x2 + (8 – 6)x – 48 = 0
⇒ x2 + 8x – 6x – 48 = 0
⇒ x(x + 8) – 6(x + 8) = 0
⇒ (x – 6)(x + 8) = 0
⇒ (x + 8) = 0 or (x – 6) = 0
⇒ x = – 8 or x = 6
A triangle's sides cannot be negative.
So, x = 6.
As a result, the right-angled triangle's base is 6 cm and its altitude is 6 + 2 = 8 cm.
As a result, the right-angled triangle's dimensions are 8 cm, 6 cm, and 10 cm.
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