find the length of each side and of the hypotenuse

Question

the hypotenuse of a right triangle is 1 foot longer than one of the sides. if the other side was 1 foot longer, it would be exactly one-third the length of the first side. find the length of each of the sides and of the hypotenuse. 

Answer 1

hypotenuse=x
one side is x-1
the third side is

 

third side =x^2-(x-1)^2,

               = sqrt (x^2-(x^2-2x+1))

                =sqrt (2x-1).

 

adding 1 to the third side makes it

 

sqrt(2x-1)+1=(1/3)(x-1)

                  =(1/3)x-(1/3)
sqrt(2x-1)=(1/3)x-(4/3)

 

square both sides

 

2x-1=(1/9)x^2-(8/9)x+(16/9)

0=(1/9)x^2-(26/9)x+(25/9)
multiply everything by 9
0=x^2-26x+25
(x-25)(x-1)=0;
x=25, 1, and 1 is impossible
x-1=24
The third side is 7 (7,24,25 triangle) and 7+1 is 1/3 of 24.