Thread: Circumcenter of a triangle

I searched through the forum and found some threads about this, but it seems like I'm doing everything write but I'm getting an unreasonable answer.

The problem: Given triangle ABC with A(1,3) B(7,-3), and C(9,5), find the circumcenter of the triangle.

Here is what i did. I found the slope of AB which is -1 so the perp slope is 1. The slope of AC is 1/4 so the perp slope is -4.

Then i found the midpt of AB and I got (4,0) and named it as point D. Next, the midpt of AC is (5,4) and I named it E.

I wasn't too sure of this step. But i set up these two formulas: y = x - 4 (i found the y-intercpt using point D), and y = 5x + 24 (using pt E).

Then I set up a system: x-4 = 5x+24 and got x=-7 and substituted it for the first formula and got y=-11. So i got (-7,-11) as the answer.

The triangle is acute and this point if completely off the triangle, and im pretty sure its wrong.

any help is appretiated

Originally Posted by majik92

[snip]

The problem: Given triangle ABC with A(1,3) B(7,-3), and C(9,5), find the circumcenter of the triangle.

[snip]

Not an answer but an interesting note: The circumcentre of the triangle is located at the centre of its circumcircle (that's the [unique] circle passing through all vertices of the triangle).

Originally Posted by majik92

I searched through the forum and found some threads about this, but it seems like I'm doing everything write but I'm getting an unreasonable answer.

The problem: Given triangle ABC with A(1,3) B(7,-3), and C(9,5), find the circumcenter of the triangle.

Here is what i did. I found the slope of AB which is -1 so the perp slope is 1. The slope of AC is 1/4 so the perp slope is -4.

Then i found the midpt of AB and I got (4,0) and named it as point D. Next, the midpt of AC is (5,4) and I named it E.

I wasn't too sure of this step. But i set up these two formulas: y = x - 4 (i found the y-intercpt using point D), and y = 5x + 24 (using pt E).

Then I set up a system: x-4 = 5x+24 and got x=-7 and substituted it for the first formula and got y=-11. So i got (-7,-11) as the answer.

The triangle is acute and this point if completely off the triangle, and im pretty sure its wrong.

any help is appretiated

Your mistake is in red. The equation of the perpendicular bisector of the line segment AC is y = -4x + 24, NOT y = 5x + 24:

Use the model y - y1 = m(x - x1), where m = -4 and a known point is (5, 4) so that x1 = 4 and y1 = 5.

Now solve x-4 = -4x+24 etc. etc.

Note that once the circumcentre is found, it makes finding the equation of the circumcircle (see previous post) quite simple.

Thank you Mr. Fantastic. I totally forgot about the y - y1 = m(x - x1) formula.

Originally Posted by majik92

Thank you Mr. Fantastic. I totally forgot about the y - y1 = m(x - x1) formula.

y = -4x + 24 is correct, but there's a simple typo in "Use the model y - y1 = m(x - x1), where m = -4 and a known point is (5, 4) so that x1 = 4 and y1 = 5." You found the typo, right ......?

At first, I was confused but I realized you meant y1 and x1. Thanks though for the rest.

this post probably comes more than a year after the actual conversation. but still, hope it helps ppl who are looking for answers.

the prop of a circumcircle is its circumference touches all three vertices of triangle. hence the circumcenter is equidistant from all the vertices of triangle. problem solved! you can make 2 linear equations from known data and solve.

The circumcenter is the intersection of the perpendicular bisectors. We need to find the midpoints of two of the sides. Next find the equation of the straight line from the midpoint to the opposite vertex. Finally, solve the two resulting linear equations.

The midpoint of BC is (8,1) and the midpoint of AC is (5,4).

One perpendicular bisector,AC, is therefore y=x-4
The other, BC, is y=(-2/7)x+21/7

I got (17/3), (5/3) for the circumcenter.