High school geometry
Features of a circle from its graph
Given a circle on the coordinate plane, Sal finds its center and its radius.
- [Voiceover] So we have a circle right over here and the first question we'll ask ourselves is what are the coordinates of the center of that circle? Well, we can eyeball that, we can see, look, it looks like the circle is centered on that point right over there and the coordinates of that point, the x-coordinate is negative four and the y-coordinate is negative seven. So the center of that circle would be the point negative four comma negative seven. Now let's say, on top of that, someone were to tell us, someone were to tell us, that this point, negative five comma negative nine is also on the circle. So negative five comma negative nine is on the circle. So based on this information, the coordinate of the center and a point that sits on the circle, can we figure out the radius? Well the radius is just the distance between the center of the circle and any point on the circle. In fact, one of the most typical definitions of a circle is all of the points that are the same distance, or that are the radius away from another point, and that other point would be the center of the circle. So how do we find out the distance between these two points, between these two points? So the length of that orange line. Well we can use the distance formula, which is essentially the Pythagorean theorem. The distance squared, so if the length of that is the distance, so we could say the distance squared is going to be equal to, is going to be equal to our change in x squared, so that right there is our change in x, and I have to write really small, but that's our change in x, our change in x squared, plus our change in y squared. Our change in y squared. Change in y squared. Now what is our change in x? Our change in x, and you can even eyeball it here, it looks like it's one, but let's verify it. It doesn't matter which one, just use the start or the end, as long as you're consistent. So, let's see, if we view this as the end, we'd say negative five it'd be negative five, minus negative four minus negative four And so this would be equal to negative one. So when you go from the center to this outer point, negative five comma negative nine, you go one back in the x direction. Now the actual distance would just be the absolute value of that, but it doesn't matter that this is a negative because we're about to square it and so that negative sign will go away. Now what is our change in y? Our change in y. Well, this is the finishing y, negative nine minus negative seven, minus our initial y is equal to negative two. And notice, just to go from that point, that y to that y, we go to negative two. So actually, we could call the length of that side as the absolute value of our change in y. And we could view this as the absolute value of our change in x. And it doesn't really matter because once we square them the negatives go away. So our distance squared, our distance squared, I really could call this the radius squared, is going to be equal to our change in x squared. Well, it's negative one squared, which is just going to be one plus our change in y squared, negative two squared is just positive four. One plus four, and so you have your distance squared is equal to five or that the distance is equal to the square root of five. And I could have just called this variable the radius, so we could say the radius is equal to the square root of five. And we're done.