Co-Ordinate Geometry (Circumcentre)

Question

When solving, is there certain points to use to get the correct answer ti find the circumcentre of a triangle with given vertex points? Say for example, point A is (0,0), point B is (4.0) and point C (1,3). I cannot solve all 3 different ways and get the same answer, so is there a certain way to remember or find out what sides to use? (ANSWER IS: (2,1))

Answer 1

You can find the intersection of any two of the perpendicular bisectors of the sides of the triangle. For this particular problem I would find equations of perpendicular bisectors of AB and AC.

The midpoint of AB is (2,0) and AB has a slope of 0 so the perpendicular through (2,0) is a vertical line with equation x = 2.

For side AC the slope is 3. Its midpoint is (1/2, 3/2). So the equation of its perpendicular bisector is:

y - 3/2 = -1/3 (x - 1/2). You can simplify that or just plug in x = 2 right now to get the y-coordinate of the circumcenter.

But if you use sides AB and BC or sides AC and BC you will get the same result. If you are not getting (2,1) then there is a mistake somewhere.

Check out:

desmos.com/calculator/ripad8h46j

This graph shows the perpendicular bisectors of all 3 sides intersecting at the same point (2,1). It also shows the circumcircle.

Answer 2

draw this on graph paper

the circumcenter is where the perpendicular bisectors of the sides intersect

you only have to find 2 of the 3 perpendicular bisectors and where they intersect is your point

the perpendicular bisector of AB is x=2 (look at the diagram on graph paper)

find the equation of CB

y=mx+b

find the slope using (1,3) and (4,0) to get -1

y=-x+b so far this line goes through (1,3) and (4,0)

substitute either one to get b=4

CB is y=-x+4

find the perpendicular bisector to CB

y=mx+b

y=x+b so far

the midpoint of CB is (2.5,1.5)

substitute to get b

b=-1

now you have y=x-1 intersecting with x=2 which is the circumcenter

so y=1 after substituting x=2

the circumcenter is (2,1)

Answer 3

You will find the equation of two of the perpendicular bisectors in point-slope form: y - y1 = m(x - x1), then determine their intersection to find the circumcenter.

Use perpendicular bisector of AC and BC (since AB has slope 0)

Find midpoints of AC and BC.

Find slopes of AC and BC.

Slope of perpendicular bisector of AC and BC will be the negative reciprocals of slopes of AC and BC.

Plug midpoints of AC and BC as (x1,y1) and slopes of perpendicular bisectors as m into point-slope formula to get equation of line in point-slope form for both.

Solve for y in both.

Equate two equations to solve for x of circumcenter, then plug back into one of the equations to solve for y of circumcenter.

Answer 4

Get the mid-points and use rt angles and intersect them.