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Math Help - Find h and k

  1. #1
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    Find h and k

    Trapezoid TEAM has the following vertices T(-2,3),
    E(1,0), A(7,6) and M(0,5). Points E, F(h,k) and A are on the same line. If T, E, F and M are the vertices of a square, find h and k.

    MY WORK:

    Since points E, F and A lie on the same line, this tells me that they have the same slope, which I found to be 1.

    This is where I got stuck.

    Where do I go from there?

    I love this coordinate geometry stuff.

    I realized that point F lies perfectly somewhere on line segment EA to form a square with the other two vertices given. Is this correct?
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  2. #2
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    Hello, magentarita!

    Please check the wording of the problem.
    As written, the facts are wrong.


    Trapezoid TEAM has vertices: . T(-2,3),\;E(1,0),\;A(7,6),\;M(0,5)

    Points E, F(h,k) and A are on the same line.
    If T, E, F, M are the vertices of a square, find h and k.
    Code:
                | 
                |                       o A
                |                 *   * 
                |           *       *
               M|     *           *
                o               *
              * | .           *
          T o   |   .       *
              * |     .   *
                *       o F
                | *   *
        - - - - + - o - - - - - - - - - -
                |   E
                |
    Quadrilateral TEFM is not a square.

    While  TM \perp TE, the length of TM is not equal to that of TE.


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    If this problem has multiple questions, okay.
    . . Otherwise, it is stupidly written.

    . . \boxed{\begin{array}{c}\text{Three vertices of a rectangle are:}\\<br />
M(0,5),\:T(\text{-}2,3),\:E(1,0)\\<br />
\text{Find }F\text{, the fourth vertex.}\end{array}}

    Who needs that trapezoid?

    Last edited by Soroban; March 10th 2009 at 04:12 PM.
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  3. #3
    A riddle wrapped in an enigma
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    [quote=magentarita;279857]Trapezoid TEAM has the following vertices T(-2,3),
    E(1,0), A(7,6) and M(0,5). Points E, F(h,k) and A are on the same line. If T, E, F and M are the vertices of a square, find h and k.


    Quote Originally Posted by Soroban View Post
    Hello, magentarita!

    Please check the wording of the problem.
    As written, the facts are wrong.

    Code:
                | 
                |                       o A
                |                 *   * 
                |           *       *
               M|     *           *
                o               *
              * | .           *
          T o   |   .       *
              * |     .   *
                *       o F
                | *   *
        - - - - + - o - - - - - - - - - -
                |   E
                |
    Quadrilateral TEFM is not a square.

    While  TM \perp TE, the length of TM is not equal to that of TE.
    Hi Magentarita,

    Soroban is absolutely right. Perhaps you meant to say rectangle instead of square.

    If you're looking to find the coordinates of F, we can procede this way.

    The slope of TM = 1. The slope of EA = the slope of EF = 1. TM is parallel to EF.

    The slope of TE is -1. Therefore the slope of MF must be -1.

    We can now set up a system of equations to solve for F(h, k).

    \text{Slope of MF}=\frac{k-5}{h-0}=-1

    k-5=-h

    [1] h=5-k

    \text{Slope of EF}=\frac{k-0}{h-1}=1

    h-1=k

    [2] h=k+1

    Solving [1] and [2],

    k+1=5-k

    \boxed{k=2}

    \boxed{h=5-2=3}

    \boxed{F(h, k)=F(3, 2)}
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  4. #4
    MHF Contributor
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    yes....

    Quote Originally Posted by Soroban View Post
    Hello, magentarita!

    Please check the wording of the problem.
    As written, the facts are wrong.

    Code:
                | 
                |                       o A
                |                 *   * 
                |           *       *
               M|     *           *
                o               *
              * | .           *
          T o   |   .       *
              * |     .   *
                *       o F
                | *   *
        - - - - + - o - - - - - - - - - -
                |   E
                |
    Quadrilateral TEFM is not a square.

    While  TM \perp TE, the length of TM is not equal to that of TE.


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    If this problem has multiple questions, okay.
    . . Otherwise, it is stupidly written.

    . . \boxed{\begin{array}{c}\text{Three vertices of a rectangle are:}\\ M(0,5),\:T(\text{-}2,3),\:E(1,0)\\
    \text{Find }F\text{, the fourth vertex.}\end{array}}" alt="
    M(0,5),\:T(\text{-}2,3),\:E(1,0)\\
    \text{Find }F\text{, the fourth vertex.}\end{array}}" />

    Who needs that trapezoid?
    I agree. The math book states that TEAM is a trapezoid but I suspected that TEFM is a rectangle and not a square. I also think the question is useless.
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  5. #5
    MHF Contributor
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    then...

    [quote=masters;279998]
    Quote Originally Posted by magentarita View Post
    Trapezoid TEAM has the following vertices T(-2,3),
    E(1,0), A(7,6) and M(0,5). Points E, F(h,k) and A are on the same line. If T, E, F and M are the vertices of a square, find h and k.




    Hi Magentarita,

    Soroban is absolutely right. Perhaps you meant to say rectangle instead of square.

    If you're looking to find the coordinates of F, we can procede this way.

    The slope of TM = 1. The slope of EA = the slope of EF = 1. TM is parallel to EF.

    The slope of TE is -1. Therefore the slope of MF must be -1.

    We can now set up a system of equations to solve for F(h, k).

    \text{Slope of MF}=\frac{k-5}{h-0}=-1

    k-5=-h

    [1] h=5-k

    \text{Slope of EF}=\frac{k-0}{h-1}=1

    h-1=k

    [2] h=k+1

    Solving [1] and [2],

    k+1=5-k

    \boxed{k=2}

    \boxed{h=5-2=3}

    \boxed{F(h, k)=F(3, 2)}
    So, what we have hidden in this geometry question is a system of linear equations in two variables h and k, right?
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