A circle with center 0,0 has radius 5 as shown in the diagram. At how many points on the perimeter of the circle are both coordinates integers? So here it is a little bit bigger. First let's notice that this circle is very rotationally symmetric, meaning if we find out, if we can count the number of points on this arc of the circle, then we can just rotate them to get all the points on the other arcs of the circle. So if we count all the points here, we can multiply by 4 to get all the points on the circle. We know that the equation of the circle is x squared plus y squared equals 25, because every point on the circle is a distance 5 away, and 5 is the square root of this guy. In order to find all the integer points, we just need to try out all of the integers. So we just need to see if all of these 5 points are integers. When x equals 5, then y equals 0, so good. When x equals 4, then y equals 3, it's the 3, 4, 5 Pythagorean triple. When x equals 3, y equals 4, so that one works as well. Now when x equals 2, y squared would have to be 21, and 21 isn't a square, so this point doesn't work. And when x equals 1, y squared would be 24, which isn't a square number, so this point doesn't work either. So the only points that stay are these 3, then by rotating it 90 degrees we get these 3 points, then by rotating it again we get these 3, and again to get the last 3. In total we have 12 points, so our final answer is 12.

circle

integer coordinates

symmetry

perimeter

points