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Thread: Geometry help needed!

  1. #1
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    Geometry help needed!

    Hi all.

    I need a help with a construction question in Geometry. I would really appreciate it if anyone could tell me how to go about it, cause I'm stumped.

    Q: Construct a triangle ABC, having given l_ ACB = 90, the hypotenuse AB = 6.4 cm, and the sum of the remaining sides AC and BC as 8 cm.

    Thanks!
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  2. #2
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    Quote Originally Posted by Insaeno View Post
    Hi all.

    I need a help with a construction question in Geometry. I would really appreciate it if anyone could tell me how to go about it, cause I'm stumped.

    Q: Construct a triangle ABC, having given l_ ACB = 90, the hypotenuse AB = 6.4 cm, and the sum of the remaining sides AC and BC as 8 cm.

    Thanks!
    Are you talking about a literal compass and straightedge construction or do you just want the lengths of the legs of the right triangle?

    Call the remaining two legs x and y.

    Then we know that
    $\displaystyle x + y = 8$
    and
    $\displaystyle x^2 + y^2 = 6.4^2$ <-- Since we have a right triangle.

    Solving the first for y gives:
    $\displaystyle y = 8 - x$

    Inserting this into the second equation gives:
    $\displaystyle x^2 + (8 - x)^2 = 48.96$

    $\displaystyle x^2 + 64 - 16x + x^2 = 48.96$

    $\displaystyle 2x^2 - 16x + 23.04 = 0$

    $\displaystyle x^2 - 8x + 11.52 = 0$

    The quadratic formula says
    $\displaystyle x = \frac{-(-8) \pm sqrt{(-8)^2 - 4(1)(11.52)}}{2 \cdot 1}$

    $\displaystyle x = \frac{8 \pm sqrt{17.92}{2}$

    $\displaystyle x \approx 1.8834~cm$ or $\displaystyle x \approx 6.1166~cm$

    So let x = 1.8834 cm, then y = 6.1166 cm. (Or x = 6.1166 cm and y = 1.8834 cm. Whichever.)

    -Dan
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  3. #3
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    Hello, Insaeno!

    This is not a simple problem . . .


    Construct a triangle ABC, having given $\displaystyle \angle ACB = 90^o$, the hypotenuse $\displaystyle AB = 6.4$ cm,
    and the sum of the remaining sides $\displaystyle AC$ and $\displaystyle BC$ is 8 cm.

    Since right triangle $\displaystyle ACB$ can be inscribed in a semicircle,
    we have this diagram:
    Code:
                    |
                  * * *   C
              *     |    o*
            *       | o   o *
           *       o|      o *
                o   |       o
          *  o      |        o*
      - A o - - - - + - - - - o B -
        -3.2                 3.2

    The circle has the equation: .$\displaystyle x^2 + y^2 \:=\:3.2^2$ . [1]

    The locus of point $\displaystyle C$ where $\displaystyle AC + BC \,= \,8$ is an ellipse.
    . . We have: .$\displaystyle a = 4$ and $\displaystyle c = 3.2$
    Then: .$\displaystyle b^2 \:=\:a^2-c^2\:=\:4^2 - 3.2^2 \:=\:5.76$
    . . The equation of the ellipse is: .$\displaystyle \frac{x^2}{16} + \frac{y^2}{5.76} \:=\:1$ . [2]

    And solve the system of equations.

    I got: .$\displaystyle x \,= \,\pm\sqrt{7},\;\;y \,= \,\pm1.8$

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  4. #4
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    Are you talking about a literal compass and straightedge construction or do you just want the lengths of the legs of the right triangle?
    I'm talking about the construction!
    I need to construct such a triangle, and thats the information given to me.
    Thanks a lot though, now that the sides are discovered the triangle can be constructed very easily.

    Soroban, I'm sorry but I didn't understand your post. I didn't understand how x can be root 7 while x + y = 8.
    I'm sorry, please clarify this.
    Last edited by Insaeno; Jun 28th 2007 at 11:04 AM.
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  5. #5
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    Quote Originally Posted by Insaeno View Post
    I'm talking about the construction!
    I need to construct such a triangle, and thats the information given to me.
    Thanks a lot though, now that the sides are discovered the triangle can be constructed very easily.

    Soroban, I'm sorry but I didn't understand your post. I didn't understand how x can be root 7 while x + y = 8.
    I'm sorry, please clarify this.
    x and y are the coordinates of the point C.

    Once you have x and y you can calculate the lengths of the sides, and
    the lengths in this case are constructible, so a construction can be
    concocted once you know what they are (though probably not worth
    the effort).

    RonL
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  6. #6
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    Hello, Insaeno!

    I didn't understand how x can be root 7 while x + y = 8.
    Read the problem again . . .

    $\displaystyle x + y \,=\,8$ is not part of the problem.
    . . This locates a point where the sum of the coordinates equals 8.

    They ask for a point whose distances from A and B has a sum of 8.

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  7. #7
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    Oh, Alright, I understand now. Thanks a lot, you've helped tremendously!!
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