Coordinates of S: (0,0) Coordinates of T: (c+b,d) Midpoint of ST: ( c+d2, d2 ) Slope of ST: dc+b
Practice makes perfect
Let's begin by determining the coordinates of S and T. Then, we can find the midpoint and the slope of ST.
Coordinates of S and T
First, notice that S is the origin. Therefore, its coordinates are (0,0). As arbitrary placeholders, let's label the coordinates of T 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
SR
d-0/c-0
d/c
RT
y-d/x-c
y-d/x-c
TQ
0-y/b-x
y/b-x
QS
0-0/b-0
0
Now we can use logic to find the values of x and y in terms of the given variables.
Since RT and QS are parallel, their slopes must be equal.
y-dx-c= 0
In order for this fraction to equal 0, the numerator must be 0.
y-d=0 ⇔ y= d
Now we know that the y -coordinate of T is d. Since SR and TQ are parallel, their slopes are also equal. Additionally, we can replace the y with d in the slope of TQ.
d/c= y/x-b ⇔ d/c= d/x-b
Because the numerators are now the same, these slopes will be equal only if the denominators are equal.
c=x-b ⇔ x= c+b
This tells us the x-coordinate of T. Finally, we have that the coordinates of point T are ( c+b, d).
Midpoint
To find the midpoint of ST, we can substitute the coordinates of S and T in the Midpoint Formula.
