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Math Help - Analytical Geometry help please!

  1. #1
    Aqua
    Guest

    Analytical Geometry help please!

    I really hope someone can help with this.

    I am afrikaans so please excuse my english names.

    We have a project in Maths asking us the following:

    In a rectangle ABC, where a line connects C directly with the basis with AB to form a 90 degrees angle(i believe its called a hightline, in direct translation).



    They now want us to prove this(Why it forms a 90 degree angle) by using Analytical geomatry, can someone help out 'n bit please?
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  2. #2
    Super Member

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    Hello, Aqua!

    I have a proof. .I hope it is what you want.


    In triangle ABC, a line connects C directly with the base AB
    to form a 90 angle.

    They now want us to prove this (why it forms a 90 angle)
    by using Analytical geometry.
    Code:
            |       C
            |       *(p,q)
            |      *: *
            |     * :   *
            |    *  :     *
            |   *   :       *
            |  *    :         *
            | *     :           *
            |*      :             *
          A * - - - * - - - - - - - * B
          (0,0)   (x,0)           (r,0)
            |       D

    Place side AC on the x-axis with vertex A at the origin.
    Let vertex B have coordinates (r,0), and vertex C (p,q).

    Draw a line from vertex V to any point D on side AC: .D(x,0).
    . . and we will minimize the length of CD.

    To make it easier, we consider the quantity, y = CD.
    . . Note that, if CD is a minimum, then CD is a minimum.

    Using the Distance Formula, we have: .y .= .(x - p) + q
    . . which simplifies to: .y .= .x - 2px + p + q

    This is an up-opening parabola; its minimum is at its vertex.
    . . The vertex of a parabola is at: .x = -b/2a

    We have: .a = 1, b = -2p. .Hence: .x .= .(-2p)/-2 .= .p

    So distance CD is a minimum when point D is (p,0).

    This means that point D is directly below point C.
    . . Hence: ./CDA = /CDB = 90

    Therefore, CD is an altitude to side AB.

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