Aladder is leaning against a wall. the top of the ladder is 9 feet above the ground. if the bottom of the ladder is moved 3 ft farther from the wall, the ladder will be lying flat on the ground, still touching the wall. how long, in feet, is the ladder?

Let L represent the ladder length, and x the distance the horiz. ladder reaches out from the wall.  Then L = x + 3, where x is the distance of the bottom of the ladder from the wall when the top of the ladder is 9 ft. above the ground.

Consider the triangle formed by the hypotenuse (L, same as ladder length), the (vertical) side opposite the angle formed by the hypo. (with length 9 ft) and the horiz side (which we will call x).  Then, according to the Pythagorean Theorem,
L^2 = x^2 + 9^2.    But L = x + 3, and L^2 = x^2 + 6x + 9 = x^2 + 9^2.  Solving this equation results in x=3.                                    6x + 9 =          9^2, or
                                                                                6x + 9 =  81
                                                                                 6x       = 72
                                                                                   x = 12
 
 But L = x+3.  So L=12+3, or   L = 15 (feet).
        

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