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
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November 2021
