In the diagram show, every point in the plane with integer coordinates is labeled with a positive integer by a “rectangular spiral” process that begins at the origin and works its way out. What are the coordinates of the point labeled 1009?
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To determine the coordinates of the point labeled 1009, we need to understand the "rectangular spiral" process. This process assigns positive integers to points in the plane with integer coordinates, starting from the origin and spiraling outward.
The spiral starts at the origin (0,0) and assigns 1. It then moves right to (1,0) and assigns 2. It then moves up to (1,1) and assigns 3. It continues moving in the spiral pattern, increasing the label by 1 each time. The pattern continues in the following order: right, up, left (twice the distance), down (twice the distance), right (three times the distance), up (three times the distance), left (four times the distance), down (four times the distance), and so on.
To determine the coordinates of the point labeled 1009, we can follow the spiral pattern until we reach 1009. We start by finding the nearest perfect square less than 1009, which is 961 (31^2). This tells us that the point labeled 1009 is somewhere on the outermost square with side length 31.
Starting at the bottom right corner of the square, we can count the number of steps needed to reach 1009. Each side of the square has a length of 31, so we need to determine how many complete sides plus extra steps we need to take.
For the first side, we move left for 30 steps (31-1) from the bottom right corner. This covers the labels from 962 to 991. Then, we move up for 31 steps, covering the labels from 992 to 1022. After that, we move right for 31 steps, covering the labels from 1023 to 1053. Finally, we move down for 31 steps, covering the labels from 1054 to 1084.
At this point, we have covered 1084 labels and need to reach 1009. We can see that we overshot our target and need to go back to the left. We move left for 25 steps (1084-1009), covering the labels from 1083 to 1059. This brings us to the point labeled 1009.
Therefore, the coordinates of the point labeled 1009 are (-25, -15).