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}

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}

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*