# Coordinate geometry

• Oct 19th 2010, 06:44 AM
Ilsa
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.
• Oct 19th 2010, 07:37 AM
Soroban
Hello, Ilsa!

Quote:

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: .$\displaystyle P(3,\,4)$
• Oct 19th 2010, 07:49 AM
Ilsa
Thanku!
It helped clear up my concepts a lot!