SOLUTION: The hypotenuse of a right angled triangle exceeds one side by 1cm and the other side by 18cm,Calculate the length of the shortest side?

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Question 967811: The hypotenuse of a right angled triangle exceeds one side by 1cm and the other side by 18cm,Calculate the length of the shortest side?
Answer by ikleyn(52781) About Me  (Show Source):
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Let  x  be the length of the hypotenuse of the given right angled triangle.

Then the length of one leg is  x-1,  and the length of the other leg is  x-18,  according to the condition.

Then the Pythagorean theorems gives an equation
%28x-1%29%5E2 + %28x-18%29%5E2 = x%5E2.

Simplify and solve it step by step to find  x.

x%5E2+-+2x+%2B+1 + x%5E2+-+36x+%2B+324 = x%5E2,

x%5E2+-+38x+%2B+325 = 0,

x%5B1%5D = %2838+%2B+sqrt%2838%5E2+-+4%2A325%29%29%2F2 = %2838+%2B+sqrt%28144%29%29%2F2 = %2838+%2B+12%29%2F2 = 25.

x%5B2%5D = %2838+-+sqrt%2838%5E2+-+4%2A325%29%29%2F2 = %2838+-+sqrt%28144%29%29%2F2 = %2838+-+12%29%2F2 = 13.

Thus there are two solutions for the hypotenuse:  25 units and/or  13 units.  Since one leg is 18 units shorter than the hypotenuse,  only one solution has sense:  x = 25.

Answer.  The hypotenuse is 25 units long.  The legs are of 24 and 7 units long.  The shortest side is of  7 units long.