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