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The Circle - Problem 4
To find the equation of a circle, you need the radius and the coordinates of the center. When given the endpoints of the diameter of a circle, the coordinates of the center will be the midpoint of the two endpoints. Use the midpoint formula to find the coordinates of the center. To find the radius, you can do one of three things: 1) Use the distance formula to find the distance between the 2 endpoints, then divide that value by 2, since radius is half the diameter; 2) Use the distance formula to find the distance between the center and one endpoint; 3) draw a triangle between the center and one endpoint and use the Pythagorean Theorem to find the distance of the diagonal, which is the radius. Once you have the radius, plus the values into the equation of a circle.
So finding a equation for a circle when we know the center and know the radius is really easy, but we're not always given that information. So sometimes we're given other information about the circle and then asked to find the equation.
In this case what we're told is the end points of a diameter and if you now remember a circle has an infinite number of diameters it's just the distance from one side of the circle to the other, so depending on where in that circle you're looking, you could have a number of different diameters.
So we're given the information about one particular diameter and what that tells us is a number of things. I'm not going to plot the points too precisely because I'm just sort of getting a general idea of what's happening.
So we know we have a point at 5,5 which is somewhere over here and we have a point at -3,1 which is somewhere over here. What we want to do is find two things; we want to find the center of the circle and we want to find the radius of the circle. We can do both of these two things using these two endpoints.
The midpoint of a diameter is going to give us the center, it's going to be a radius on either sides so therefore that will the center. If you remember how to find the midpoint of a segment, it's just the average of the xs and average of the ys, so let's do that first.
We know that the center then is just going to be the average of your x values which is just going to be 5 plus -3 over 2 and 5 plus -1 over 2. Calculating this out, 5 plus -3 is 2, 2 over 2 is 1. 5 plus -1 is 4, 4 over 2 is 2, so we have found the center to be the point 1,2 and this point over here is 5,5.
So now we have the center, half the battle. We also need to find the radius and we have a couple ways of finding the radius. I'll tell you about a couple of them and then we'll jut do end up doing one of them. The ways we can find the radius, first what we can do is take the distance between our two end points using the distance formula and that will give us the diameter. Diameter divided by 2 is going to give us the radius, one way.
We could also use the distance formula between these two points. Again using the distance formula, that would just give us the radius straight away. The other way which is a little bit trickier is to just draw a triangle using these two points on a coordinate and axis, so all we have to do is draw in this triangle and then we can use the Pythagorean Theorem because we know the difference between our x coordinate and our y coordinates.
So looking at our x coordinates, we go from 1 to 5, so the base of this triangle is a distance of 4. Looking at our y coordinates, we go from 2 to 5 so our y is a distance of 3 and that leaves us with a 3, 4, 5 triangle. This has to be a right angle because we're dealing with change of x's plus change of y's, so we can just look and know that our radius has to be 5.
So what we've done is we have evaluated some information using our two endpoints, we found our midpoint which is going to be the center and we found our radius using the Pythagorean Theorem. Now all we have to do is plug this into our circle equation which we just know is the x minus the x coordinate on the center, just 1, plus y minus the y coordinate of the center which is 2 those both squared equals radius squared of 25.
We're a little bit more involved than just being given the center and the radius, but still very attainable.