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Math Help - Interesting Euclidean Geoboard Problem

  1. #1
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    Interesting Euclidean Geoboard Problem

    I ran into this exercise a couple of days ago and originally thought it had no business in the collegiate level textbook where I found it because it was so simple. (notice that two of the vertices are not on a known point).

    The exercise asks:

    a) find the area of the quadrilateral
    b) prove your solution

    Given: The vertical and horizontal distance between any two dots (pegs in geoboard speak) is 1 unit. (As you probably suspected).


    [IMG]file:///C:/Users/Terry/AppData/Local/Temp/moz-screenshot-48.png[/IMG]Interesting Euclidean Geoboard Problem-geoboard-exercise-3.1.jpg I was able to find the area using trig and a fair bit of algebra. However, I haven't been able to find an applicable Euclidean theorem to do the trick. My instincts tell me to apply the mid-segment theorem, but that idea hasn't done much to solve the exercise.

    Thanks for reading!
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  2. #2
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    There is an amazing formula for calculating the area of any polygon from its vertex coordinates.
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  3. #3
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    Thanks Emakarov! That is a great formula. Unfortunately it doesn't work with this problem because without trig, i can't say what two of the coordinates are.
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  4. #4
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    You only need elementary algebra to find the coordinates of the vertices. Suppose this is a coordinate plane with (0, 0) in the lower-left corner. Then the equation of the line that goes through (0, 0) and (2, 1) is y = x/2. The equation of the line that goes through (3, 2) and (4, 4) is y - 2 = 2(x - 3), i.e., y = 2x - 4. Solving the system y = x/2 and y = 2x - 4 gives x = 2 2/3 and y = 1 1/3. From symmetry, the opposite vertex has x and y coordinates reversed.
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  5. #5
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    Another formula for the area of rhombus is half the product of the diagonals. Knowing the vertices, you can find the diagonals.

    Still another way is to find the altitude from (4, 4) to the line y = x/2. It is well-known that y = ax + b and y = cx + d are perpendicular iff ac = -1. So the equation of the altitude is y = -2x + 12. Now you can find its intersection with y = x/2 and thus find the length of the altitude.
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  6. #6
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    Cool!

    You are just a wealth of cool formulas, aren't ya?

    Actually, I solved the area in a bunch of ways, but i finally proved it using the median of a triangle theorem. The solution i came up with is attached.

    Thanks again!
    Attached Thumbnails Attached Thumbnails Interesting Euclidean Geoboard Problem-geometry-curiosity-32-ex-20-proven.pdf  
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