How to Find the Points of a Circle
Points of a Circle
Points on a circle. A circle in the coordinate plane has a center at (3,1). One point on the circle is (6,-3). Name three more points on the circle. The equation of a circle is X minus H squared plus Y minus K squared is equal to R squared.
This might look familiar to you because it’s derived from the distance formula. If you take the square root of both sides, the square root of R squared would just be R and the square root of X minus H squared plus Y minus K squared is the formula for finding the distance between two points.
Which is what a radius is. A radius is the distance between the center of the circle and the outside or a point on the outside of a circle. The H and the K are the center points. Your center is at (H,K) and then the X and Y are just any other point that you might have.
R of course is the radius, stands for radius. We can use the information we’ve been given to find our radius. Since they told us what our center was, the center is (3,1), and they gave us another coordinate which we could plug in for our X and our Y. We have coordinates to substitute for X,Y,H, and K and that would leave only R or radius, uknown. Well, If we know our radius, and we know our center, then we can find many other points on our circle.
We’re going to start by finding our radius by plugging in our point for our center and for our other coordinate. Since this is my centerpoint, this is my H and my K, and then this will be my X and Y. X is 6 minus H is 3, squared, plus Y is -3 minus K is 1, squared.
That’s equal to R squared. To find R we need to simplify everything on the left side. 6 minus 3 squared. 6 minus 3 is 3 squared, plus -3 minus 1, so it’s plus a negative. -3 plus negative 1 would be -4, squared, equals radius squared.
3 squared is 9, plus -4 squared is 16 equals R-squared. 9 plus 16 is 25, so 25 equals your radius squared. To find the radius we need to get rid of our squared or get radius alone. The opposite of squaring number is to square root.
We need to square root both sides. The square root of R squared is R and the square root of 25 is 5. That means that our radius is 5. I’m going to use this coordinate plane to graph our center point (3,1) over 1, 2, 3. Up one.
Then I’m going to use the radius to find three other points that would be on our circle. (6,-3) is one they already gave us. (6,-3). From this point, the radius goes 5 units away in any direction and the easiest units to count would be vertical units and horizontal units.
So I’m going to use those. So from (3,1) we need to go up 1, 2, 3, 4, 5. There is a point on our circle that’s over 3 up 6, or we could go to the left. 1, 2, 3, 4, 5, so that is (-2,1). From our center we could go down 5. 1, 2, 3, 4, 5.
Which is (3,-4). Those are three points that are on our circle. You can maybe start to see it forming. If we started connecting our points, we could get a circle. You could find lots of other points using pythagorean theorem.
You can think of the radius like your hypotenuse, so the hypotenuse is 5 of your right triangle, and a 3 4 5 right triangle is one of our pythagorean triples. We could go from the center, we could go up 1, 2, 3 and over 1, 2, 3, 4 for another point on our circle.
We could do that in any direction. But, here are three points that are on our circle besides the one that was given.