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Math Help - Finding the lengths of a rectangle (not as easy as it sounds)

  1. #1
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    Finding the lengths of a rectangle (not as easy as it sounds)

    I have a quiz coming up in my Geometry class and my teacher has hinted that there will be a question like this one on the quiz:

    "Ashley saved a distance equal to 80% of the length of the shortest side of a rectangular field by cutting across the diagonal of the field instead of along two sides. Find the ratio of the length of the shortest side of the field to the length of its longest side."

    I tried working on it with a friend of mine but we don't even know how to set it up (much to my embarrassment.) I need help to solve this question. Please help me ASAP because I believe the quiz will be in two days. Any info that can help will work.

    Thanks.
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  2. #2
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    Re: Finding the lengths of a rectangle (not as easy as it sounds)

    I'm afraid this question as written doesn't make any sense. How can the diagonal be only 20% of the length of the shortest side, when the diagonal is the hypotenuse of a right angle triangle made by the lengths of the rectangle, and the hypotenuse is the longest side?
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    Re: Finding the lengths of a rectangle (not as easy as it sounds)

    I believe the question means that Ashley saved 80% of the shortest side as in (i.e.) the shortest side is 10 ft. meaning she saved 8 feet less by walking on the diagonal rather than along two of the sides. Does that help clear it up? *crossing fingers*
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    Re: Finding the lengths of a rectangle (not as easy as it sounds)

    Let x and y be the lengths of the sides of the rectangle. The distance across the diagonal is  \sqrt {x^2+y^2} which equals the length around x + y less 80% of y. or:

     \sqrt {x^2+y^2} = x + 0.2y

    Square both sides and simplify:

     x^2 + y^2 = x^2 + 0.4xy + 0.04y^2

     0.96y^2 = -.4xy

     0.96 y = -.4 x

    x/y =  2.4
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    Re: Finding the lengths of a rectangle (not as easy as it sounds)

    Oh! I get what you're saying. Question though. Doesn't 0.4xy remain positive in the third step of solving the solution? x/y would be 2.4 but how would I write a ratio of the length of the shortest side of the field to the length of its longest side?
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    Re: Finding the lengths of a rectangle (not as easy as it sounds)

    Quote Originally Posted by MyHappyHarmony View Post
    Oh! I get what you're saying. Question though. Doesn't 0.4xy remain positive in the third step of solving the solution? x/y would be 2.4 but how would I write a ratio of the length of the shortest side of the field to the length of its longest side?
    Already given. x/y is the ratio, which in decimals is 2.4. Make that into fractions, you have 12 over 5. Change to ratio form and you have 5:12.
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    Re: Finding the lengths of a rectangle (not as easy as it sounds)

    Understood. Thanks so much to you, ebaines, and Prove It!
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