Find the missing leg of the triangle

A garden has the shape of a right triangle with one leg 9 meters longer than the other. The hypotenuse is 9 meters less than twice the length of the shorter leg. What is the length of the shorter leg?

x = one leg
x - 9 = other leg {one leg is 9 shorter than the other}
2(x - 9) - 9 = hypotenuse {the hypotenuse is 9 less than twice the length of the  shorter leg}

a² + b² = c²  {the Pythagorean Theorem}
a and b are the legs and c is the hypotenuse

x² + (x - 9)² = [2(x - 9) - 9]²   {substituted into the Pythagorean Theorem}
x² + (x - 9)² = [(2x - 18) - 9]²  {used distributive property}
x² + (x - 9)² = (2x - 27)²   {combined like terms in brackets}
x² + x² - 18x +  81 = 4x² -108x + 729  {used foil method to square the binomials}
2x² - 18x + 81 = 4x² - 108x + 729   {combined like terms}
2x² - 90x + 648 = 0   {subtracted 2x² and 81 and added 18x to each side}
x² - 45x + 324 = 0   {divided each side by 2}
(x - 36)(x - 9) = 0   {factored into two binomials}
x - 36 = 0 or x  - 9 = 0   {set each factor equal to 0}
x = 36 or x = 9   {solved each equation for x}
x = 36 is the only solution that will work

x = 36 {the longer leg}
x - 9 = 27 {the shorter leg}
2(x - 9) - 9 = 45 {the hypotenuse}

27 meters is the length of the shorter leg

- Algebra House