How to Find the Circumcenter of a Triangle

This video shows how to find the circumcenter of a triangle. The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect. To find it, you need to find the point where the perpendicular bisectors of at least two sides meet. You only need two because the third perpendicular bisector will meet at the same point.

Circumcenter of a Triangle

Circumcenter of a Triangle

The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect. Perpendicular meaning “at right angles,” and bisect which means “cuts in half.” To find the circumcenter of a triangle, you would need to find the point where the perpendicular bisectors of a least 2 of the sides meet.

You only need to find where 2 meet because the third perpendicular bisector will meet at the same point. If you find the point where 2 of the perpendicular by sectors meet, the third will meet at the same point, so you don’t need it to find the circumcenter. The circumcenter is the center of a circle that could be circumscribed about the triangle.

Let’s look at an example: What is the circumcenter of a triangle with vertices A (negative 3, 0), B (1, 4), and C (5, negative 2)? As I said just a minute ago, you only need to find the perpendicular bisectors of 2 sides, and it doesn’t matter which 2 sides you use, so I’m going to start with segment A, B.

The first thing I’m going to find on segment A, B is the mid-point, since a perpendicular bisector cuts my line in half (so right in the middle). I need to find where my perpendicular line is going to be drawn, which means I need to find the middle of segment A, B. Which means we’re going to use the midpoint formula, which is: add your Xs, divide by 2, then add the Ys, and divide by 2.

Since I’m finding the midpoint of segment A, B, this will be my (X_1, Y_1), and B will be my (X_2, Y_2) (and you could do it in the complete reverse, it doesn’t matter, but, since this was given to me first, I’ll use it as my first and that as my second). My X values are negative 3, and 1, negative 3 plus 1, divided by 2, and then my Y values are 0 and 4, so 0 plus 4, divided by 2.

Then we need to simplify, so the midpoint of segment A, B is negative 3 plus 1, negative 2, divided by 2, 0 plus 4 is 4, divided by 2. Then we need to simplify one more time. The midpoint of the segment A, B is negative 2 divided by 2, negative 1, and 4 divided by 2, 2.

Now I know where the middle of segment A, B is, and that’s at (negative 1, 2), (so negative 1, up 2). and that’s where my perpendicular line is going to go through. I could find the middle of another segment now, or I could continue on with segment A, B and find the perpendicular line (or find the slope of the line) that’s going to go through that point.

I’m going to do that, which means I need to know the slope of line A, B to find out what the slope of my perpendicular line will be. The slope formula is Y_2 minus Y_1, divided by X_2 minus X_1. I’m going to find the slope of line A, B. I’m using the same point, so I have the same (X_1, Y_1), (X_2, Y_2). Y_2 then would be 4, minus Y_1 is 0, divided by X_2 is 1, minus X_1 is negative 3.

Then we need to simplify, 4 minus 0 is 4, divided by 1 minus negative 3 (add the inverse) 1 plus 3 is 4, so it’s 4 divided by 4, which is 1. The slope of line A, B is 1 (up 1, over 1, up 1, over 1, up 1, over 1). My perpendicular line has an opposite reciprocal slope, so since the slope of line A, B is 1, that means that my perpendicular slope is negative 1.

The slope of my perpendicular line going through this point it’s going to be negative 1. Okay before I get to drawing any lines, I’m going to go ahead and do the same thing on my other line, and I’m going to use segment A, C, although you could use segment B, C.

I’m going to do the same thing we just did, the midpoint of segment A, C is—I’m using A and C now, so this will be my (X_2, Y_2)—so I need to add my Xs from A and C, negative 3 plus 5, divided by 2, and then add my Ys from A and C, 0 and negative 2, and then we simplify.

Negative 3 plus 5 is 2, divided by 2, 0 plus negative 2 is negative 2, divided by 2. The midpoint of segment A, C is 2 divided by 2 is 1, negative 2 divided by 2 is negative 1. The middle of segment A, C is (1, negative 1) (1, negative 1). Now I’m going to find the slope of line A, C, so that I can find the slope of the perpendicular bisector for segment A, C.

Again, I’m finding slope using the same slope formula: Y_2 minus Y_1, divided by X_2 minus X_1—change in Y divided by change in X—and I’m using for A, C now (A and C). Y_2 is negative 2, minus Y_1 is 0, divided by X_2, 5, minus X_1, negative 3, and then we simplify.

The slope of segment A, C is: negative 2 minus 0 is negative 2, divided by 5 minus a negative 3 (plus a positive three) would be 8, and that simplifies to negative 1/4. The slope of line A, C is negative 1/4, and my perpendicular line has an opposite reciprocal slope, so the perpendicular slope would be positive 4 over 1, or just positive 4.

Now that we have this information, we can find the equations of the perpendicular bisector lines, and once we have those equations, we can see at what point those lines actually intersect. That’s what I’m going to do now, I’m going to find the equation of the line for the perpendicular bisector of segment A, B.

We have this information: the midpoint of segment A, B is (negative 1, 2), and the perpendicular slope for that line was negative 1—the perpendicular bisector of segment A, B passes through the point (negative 1, 2) and has a slope of negative 1. I can use that information to plug into point-slope form, which is Y minus Y_1 equals the slope, times the quantity, X minus X_1—this is my (X_1, Y_1).

Y minus, Y_1 is 2, equals, the slope is negative 1, times the quantity, X minus, X is negative 1. Here we can add the inverse, and then I’m going to simplify my equation. This is Y minus 2 is equal to negative X, minus 1 (and just distributed the negative 1). Then add 2 to both sides, and Y is equal to negative X, plus 1.

That’s the equation for the perpendicular bisector of segment A, B. I’m going to do the same thing for segment A, C. For segment A, C the midpoint was (1, negative 1), and the perpendicular slope was 4. Again, I have a point and a slope and I’m going to substitute into point slope form. Y minus Y_1 is equal to the slope, times the quantity, X minus X_1.

Again, this is my (X_1, Y_1). Y minus, Y_1 is negative 1, (so add the inverse) is equal to, the slope is 4, times the quantity, X minus X, which is 1, and then we simplify. Y plus 1 is equal to, (distribute the 4) 4X minus 4, then subtract 1 from both sides, so the equation is Y equals 4X, minus 5. Whew, Okay.

Now, we take those 2 equations—those are, again, the equations of the lines of the perpendicular bisectors of these 2 sides A, B and A, C—and we need to find where they intersect, because where they intersect is my circumcenter. To find where they intersect, I’m just going to substitute 4X minus 5 for Y, so I have 4X minus 5 is equal to negative X plus 1.

Again, all I did to come up with that was substitute 4X minus 5 for Y, since Y equals 4X minus 5. Then solve for X. Add X to both sides, 4X plus X is 5X, minus 5, is equal to, negative X plus X cancels, 1. Add 5 to both sides, 5X is equal to 6. Then divide the sides by 5 to solve for X. X is 6/5, or if you wanted to change into a mixed number, 5 goes into 6, 1 time, with 1 left over out of 5.

I’m halfway there I know what my X-value is. Now I need to find my Y-coordinate. You can use either one of these equations to find your Y-coordinate (this one looks easier, so that’s the one I’m going to use). Y is equal to the opposite of X, and X is 1 and 1/5, plus 1. Negative 1 and 1/5, plus 1, is a negative 1/5.

Now that I know my X- and my Y-coordinate, I just put those together for my circumcenter. My circumcenter is 1 and 1/5, and negative 1/5 (over 1 and 1/5, and down a negative 1/5). About right there would be the center of a circle that you could circumscribe about your triangle, and that’s our answer.

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by Mometrix Test Preparation | Last Updated: August 15, 2019