help please
The measures of the sides of a right triangle form an arithmetic progression whose difference is 1. Find the measure of the hypotenuse
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
__________________________________
Remark:
( a + b ) ^ 2 = a ^ 2 + 2 * a * b + b ^ 2
__________________________________
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