The hypotenuse of a right triangle is 13 ft long. The longer leg is 7 ft longer than the shorter leg. Find the side lengths of the triangle.

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The hypotenuse of a right triangle is 13 ft long. The longer leg is 7 ft longer than the shorter leg. Find the side lengths of the triangle.

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  1. Answer:

    Step-by-step explanation:

    Let x be the longer leg.

    Then the shorter leg is (x-7) centimeters long.

    According to the Pythagorean theorem,

       x%5E2 + %28x-7%29%5E2 = 13%5E2,   or

       x%5E2+%2B+x%5E2+-+14x+%2B+49 = 169

       2x%5E2+-+14x+-+120 = 0

       x%5E2+-+7x+%2B+60 = 0

    Factor the quadratic polynomial

       (x-12)*(x+5) = 0.

    Its roots are x= 12 and  x= -5,  but only positive solution is meaningful.

    ANSWER.  The longer leg is 12 cm;  the shorter leg is  12-7 = 5 cm.

    CHECK.   12%5E2 + 5%5E2 = 144 + 25 = 169 = 13%5E2.    ! Correct !

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  2. Answer:

    The shorter leg is five feet, the longer leg is 12 feet, and the hypotenuse is 13 feet.

    Step-by-step explanation:

    Let the shorter leg be x.

    Since the longer leg is seven feet longer than the shorter leg, the length of the longer leg can be modeled by (x + 7).

    Since the triangle is a right triangle, we can use the Pythagorean Theorem, given by:

    [tex]a^2+b^2=c^2[/tex]

    Where a and b are the side lengths and c is the hypotenuse.

    The hypotenuse is 13 and the legs are x and (x + 7). Substitute:

    [tex](x)^2+(x+7)^2=(13)^2[/tex]

    Square:

    [tex]x^2+x^2+14x+49=169[/tex]

    Simplify:

    [tex]2x^2+14x-120=0[/tex]

    We can divide both sides by two:

    [tex]x^2+7x-60=0[/tex]

    Factor:

    [tex](x-5)(x+12)=0[/tex]

    Zero Product Property:

    [tex]x-5=0\text{ or }x+12=0[/tex]

    Solve for each case:

    [tex]x=5\text{ or } x=-12[/tex]

    Since lengths cannot be negative, we can ignore the negative answer. So, our only solution is:

    [tex]x=5[/tex]

    The shorter leg is five feet, the longer leg will be (5 + 7) or 12 feet. And the hypotenuse is 13 feet as given.

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