Um, i need help understanding this equation?

Question

The hypotenuse of a right triangle is 10 inches. The length of the two legs are given by two consecutive even integers. Find the length of the two legs.

 

The lengths of the two legs are _______ and ______ inches 

Answer 1

you should be able to guess because of knowledge of the 3-4-5 Pythagorean triple.  A triangle 6-8-10 (all lengths of 3-4-5 triangle doubled) is also a right triangle and its legs happen to be consecutive even integers.

 

Algebraically, solve 10² = a² + (a+2)².

Answer 2

Let x be the smaller of the two unknown integers.  Since they are consecutive even integers, the larger integer is x+2.  Now use the Pythagorean Theorem (a² + b² = c²):

 

     x² + (x+2)² = 10²

     x² + x² + 4x + 4 = 100

     2x² + 4x - 96 = 0

     2x² + 16x - 12x - 96 = 0

     2x(x+8) - 12(x+8) = 0

     (2x-12)(x+8) = 0

 

x = 6 and -8.  Since we can't have a negative length, x = 6 inches.  The longer leg is x+2 = 8 inches.  Check:

 

     6² + 8² = 10²

     36 + 64 = 100

     100 = 100

     Check!

 

(The right triangle 6-8-10 is a member of the 3-4-5 family of Pythagorean Triples)