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
