Let A0=(−1,0), and let O be the origin (0,0). For each integer i≥1, we construct the point Ai so that |Ai−1Ai|=|Ai−1O| and the angle ∠OAi−1Ai is a right angle. If O,Ai−1,Ai+1 are not collinear for any value of i, what is the x-coordinate of A12?
To find the x-coordinate of A12, we need to go through the construction steps for each Ai.
Step 1: Start with A0 = (-1, 0) and O = (0, 0).
Step 2: To find A1, we construct a line segment Ai-1Ai such that its length is equal to the distance between Ai-1 and O. Since |Ai-1O| = |-1 - 0| = 1, A1 will be 1 unit to the right of A0. So A1 = (0, 0).
Step 3: To find A2, we construct a line segment Ai-1Ai such that its length is equal to the distance between Ai-1 and O. Since |Ai-1O| = |0 - 0| = 0, A2 will be 0 units to the right of A1, which means it will be at the same position as A1. So A2 = (0, 0).
Step 4: Continuing with the pattern, we find that for each subsequent Ai-1 and Ai, the distance between Ai-1 and Ai will be the same as the distance between Ai-1 and O. Therefore, A3, A4, A5, and so on will all have the same coordinates as A2, which is (0, 0).
Since all the Ai for i ≥ 2 have the same x-coordinate of 0, the x-coordinate of A12 will also be 0.
Therefore, the x-coordinate of A12 is 0.