1. ## 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.

2. 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: .$\displaystyle P(3,\,4)$

3. Thanku!
It helped clear up my concepts a lot!