4)
calculate the midpoint of AB and AC:
AB/2: (1.5,0)
AC/2: (1.5,1)
calculate the slopes m for AB and AC:
AB: m=(0-0)/(3-0)=0/3=0
AC: m=(2-0)/(3-0)=2/3
calculate the perpendicular slopes/lines
for AB: AB is horizontal, so the perpendicular one has to be vertical
so instead of the formula y=0 for the AB line, the line through the midpoint AB/2 is
x=1.5
for AC:
m1*m2=-1
m1=-1/m2
m1=-1/(2/3)
m1=3/-2=-2/3
-> y=-2/3x+d
insert AC/2 to calculate d:
1=-2/3*(3/2)+d
1=-1+d
2=d
so it is y=-2/3x+2
find the intersection of both lines by substitution of x=1.5:
y=-2/3x+2
insert x=1.5
y=-2/3*(3/2)+2
y=-1+2
y=1
so (1.5,1) is the the circumcenter
5)
in essence you can do the same calculation or take shortcuts to verify the possible solutions:
a simple on is if two vertices are on the same height/length, then the cirumcenter coordinate for the other axis is at the midpoint for those vertices
in this case B and C are on (*,2), so they share their y height
this means the x coordinate of the circumcenter is the midpoint of both them:
mid of 1 and 6=5/2+1=2.5+1=3.5
so the circumcenter is at (3.5,?) only one solution matches this: (3.5,3) so the 4th answer is the solution
