We want to determine the coordinates of P. We will do that by first determining the coordinates of O. Then we will use that for a parallelogram both pairs of opposite sides are parallel.
Coordinates of P
First, notice that O is the origin. Therefore, its coordinates are (0,0). As arbitrary placeholders, let's label the coordinates of P as (x,y). We can call the other two points R and Q.
We know that the given figure is a parallelogram, so both pairs of opposite sides are parallel. Let's find the slopes of the sides by substituting the coordinates of the points into the Slope Formula.
Side
Slope Formula
Simplify
QR
c-0/- b-(- a)
c/- b+a
RP
y-c/x-(- b)
y-c/x+b
PO
0-y/0-x
y/x
OQ
0-0/- a-0
0
Now we can use logic to find the values of x and y in terms of the given variables.
Since RP and OQ are parallel, their slopes must be equal.
y-cx+b= 0
In order for this fraction to equal 0, the numerator must be 0.
y-c=0 ⇔ y= c
Now we know that the y -coordinate of P is c. Since QR and PO are parallel, their slopes are also equal. Additionally, we can replace the y with c in the slope of PO.
c/- b+a= y/x ⇔ c/- b+a= c/x
Because the numerators are now the same, these slopes will be equal only if the denominators are equal.
- b+a=x ⇔ x= a-b
This tells us the x-coordinate of P. Finally, we have that the coordinates of point P are ( a-b, c).
