Answer:
A) The coordinates of the fourth vertex are:
1) x-coordinate:
[tex]x=2[/tex]
2) y-coordinate:
[tex]y=-1[/tex]
B) The point of intersection of the diagonals is: [tex](2,1)[/tex]
Step-by-step explanation:
We need to remember that the diagonals of a parallelogram intersect each other at a half-way point and the midpoint of each diagonal is the same.
The midpoint formula is:
[tex]M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]
Since:
[tex]M_{AC}=M_{BD}[/tex]
We can find the coordinates of the fourth vertex [tex]D(x,y)[/tex] through these procedure:
1) x-coordinate:
[tex]\frac{1+3}{2}=\frac{2+x}{2}\\\\2(2)=x\\\\4-2=x\\\\x=2[/tex]
2) y-coordinate:
[tex]\frac{0+2}{2}=\frac{3+y}{2}\\\\1(2)-3=y\\\\y=-1[/tex]
Therefore, fourth vertex is [tex]D(2,-1)[/tex]
Since the point of intersection of the diagonals is the midpoint of a diagonal (Remember that [tex]M_{AC}=M_{BD}[/tex]), this is:
[tex]M_{AC}=M_{BD}=(\frac{1+3}{2},\frac{0+2}{2})\\\\M_{AC}=M_{BD}=(2,1)[/tex]
Therefore, the point of intersection of the diagonals is [tex](2,1)[/tex]
