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Math Help - Lines

  1. #1
    Junior Member Dragon's Avatar
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    Lines

    Consider the line L containg the points (-3,8)and (6,-4) what is the length of the hypotenuse of the right triangle formed b the intersection of L and the x and Y axes
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  2. #2
    Super Member

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    Hello, Dragon!

    Did you make a sketch?


    Consider the line L containg the points (-3,8)and (6,-4).
    What is the length of the hypotenuse of the right triangle
    formed by the intersection of L and the x- and y-axes?
    Code:
      *    (-3,8)       |
            *           |
                  *    Q|
                        *
                        |     *      P
      - - - - - - - - - + - - - - - * - - - - - - -
                        |                 *     (6,-4)
                        |                       *
                        |                            *

    Game plan
    The line L contains the points (-3,8) and (6,-4).
    . . We want the equation of that line.
    Then we want its x-intercept P and y-intercept Q.
    . . Then we want the distance \overline{PQ}.

    The slope of line L is: . m \:=\:\frac{-4 - 8}{6 -(-3)} \:=\:\frac{-12}{9}\:=\:-\frac{4}{3}

    The equation of the line through (6,-4) with slope -\frac{4}{3} is:
    . . y - (-4)\:=\:-\frac{4}{3}(x - 6)\quad\Rightarrow\quad y\:=\;-\frac{4}{3}x + 4


    For the x-intercept, let y = 0 and solve for x.
    . . 0 \:=\:-\frac{4}{3}x + 4\quad\Rightarrow\quad x = 3 . . . x-intercept: P(3,0)

    For the y-intercept, let x = 0 and solve for y.
    . . y\:=\:-\frac{4}{3}\cdot0 + 4\quad\Rightarrow\quad y = 4 . . . y-intercept: Q(0,4)


    The distance from P(3,0) to Q(0,4) is:

    . . PQ\:=\:\sqrt{(0-3)^2 + (4-0)^2} \;=\;\sqrt{9+16}\;=\;\sqrt{25}\;=\;\boxed{5}

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