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Thread: Coordinate geometry

  1. #1
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    Exclamation Coordinate geometry

    Three points have coordinates o(0,0), A(5,0) and B(7,6). If P is the point (x,y) where y>0, calculate the value of x and y given that AP=BP and that the area of the triangle AOP is 10sq. units.
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  2. #2
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    Hello, Ilsa!

    Three points have coordinates: .$\displaystyle O(0,0),\: A(5,0),\:B(7,6)$

    If $\displaystyle \,P(x,y)$ where $\displaystyle y>0$, calculate the value of $\displaystyle \,x$ and. $\displaystyle \,y$

    given that $\displaystyle AP=BP$ and that the area of $\displaystyle \Delta AOP =10 \text{ units}&^2$
    Code:
            |             (x,y)           (7,6) 
            |             P o               * B
            |                  o        *  *
            |                     o *     *
            |                   *    o   *        
            |               *           o
            |           *              *   o
            |       *                 *       o
            |   *                    *
        - - * - - - - - - - - - - - * - - - - - -
          (0,0)                   (5,0)
            O                       A

    Since $\displaystyle \,P$ is equidistant from $\displaystyle \,A$ and $\displaystyle \,B$
    . . $\displaystyle \,P$ lies on the perpendicular bisector of $\displaystyle \,AB,$
    . . which has the equation: .$\displaystyle y \:=\:-\frac{1}{3}x + 5$



    Code:
            |               P 
            |               o (x,y)
            |             * :*
            |           *   : *
            |         *     :  *        
            |       *       :y  *
            |     *         :    *
            |   *           :     *
           O| *             :      * A
        - - * - - - - - - - + - - - * - -
          (0,0)                   (5,0)

    The area of $\displaystyle \Delta AOP$ is 10 units$\displaystyle ^2.$

    Its base is 5 and its height is $\displaystyle \,y$.
    . . $\displaystyle A \,=\,\frac{1}{2}bh \quad\Rightarrow\quad \frac{1}{2}(5)y \:=\:10 \quad\Rightarrow\quad y \,=\,4$

    Hence: .$\displaystyle -\frac{1}{3}x + 5 \:=\:4 \quad\Rightarrow\quad x \,=\,3$


    Therefore: .$\displaystyle P(3,\,4)$
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  3. #3
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    Thanku!
    It helped clear up my concepts a lot!
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