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Mathematics

The hypotenuse of a right-angled triangle exceeds one side by 1 cm and the other side by 18 cm; find the lengths of the sides of the triangle.

Quadratic Equations

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Answer

Let hypotenuse of a right-angled triangle = x cm

So, other sides are (x - 1) cm and (x - 18) cm.

⇒ (Hypotenuse)2 = (Perpendicular)2 + (Base)2

⇒ (x)2 = (x - 1)2 + (x - 18)2

⇒ x2 = x2 + 1 - 2x + x2 + 324 - 36x

⇒ x2 = 2x2 - 38x + 325

⇒ 2x2 - x2 - 38x + 325 = 0

⇒ x2 - 38x + 325 = 0

⇒ x2 - 25x - 13x + 325 = 0

⇒ x(x - 25) - 13(x - 25) = 0

⇒ (x - 13)(x - 25) = 0

⇒ x - 13 = 0 or x - 25 = 0

⇒ x = 13 or x = 25.

Since, one side is 18 cm less than hypotenuse so,

∴ x ≠ 13 as x - 18 = -5 and side cannot be negative.

∴ x = 25, x - 1 = 24 and x - 18 = 7.

Hence, length of the sides of triangle = 7 cm, 24 cm and 25 cm.

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