help please

Question

The measures of the sides of a right triangle form an arithmetic progression whose difference is 1. Find the measure of the hypotenuse

Answer 1

a = first leg

b = a + 1 = second leg

c = b + 1 = a + 1 + 1 = a + 2 = hypotenuse

c ^ 2 = a ^ 2 + b ^ 2

( a + 2 ) ^ 2 = a ^ 2 + ( a + 1 ) ^ 2

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Remark:

( a + b ) ^ 2 = a ^ 2 + 2 * a * b + b ^ 2
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a ^ 2 + 2 * a * 2 + 2 ^ 2 = a ^ 2 + a ^ 2 + 2 * a * 1 + 1 ^ 2

a ^ 2 + 4 a + 4 = a ^ 2 + a ^ 2 + 2 a + 1

a ^ 2 + 4 a + 4 = 2 a ^ 2 + 2 a + 1 Subtract a ^ 2 + 4 a + 4 to both sides

a ^ 2 + 4 a + 4 - ( a ^ 2 + 4 a + 4 ) = 2 a ^ 2 + 2 a + 1 - ( a ^ 2 + 4 a + 4 )

0 = 2 a ^ 2 + 2 a + 1 - a ^ 2 - 4 a - 4

0 = a ^ 2 - 2 a - 3

a ^ 2 - 2 a - 3 = 0

The solutions are :

x = - 1

and

x = 3

Leg can not be negative so :

a = 3 , b = 3 + 1 = 4 , c = 4 + 1 = 5