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 understand the pattern and construct each point Ai step by step.
Given A0 = (-1, 0) and O = (0, 0), we are asked to find the x-coordinate of A12.
Step 1:
To find A1, we need to construct a point with the same distance from A0 as the distance between A0 and O and form a right angle with OA0.
Since A0 = (-1, 0) and O = (0, 0), the distance between A0 and O is 1 unit. A1 can be found by moving 1 unit from A0 towards O in the same x-direction. Hence, A1 = (-2, 0).
Step 2:
To find A2, we need to construct a point with the same distance from A1 as the distance between A1 and O and form a right angle with OA1.
The distance between A1 and O is 1 unit. A2 can be found by moving 1 unit from A1 towards O in the same x-direction. Hence, A2 = (-3, 0).
Step 3:
Using the same logic as in Step 2, we continue constructing each point Ai using the distances between Ai-1 and O.
To find A3, we move 1 unit from A2 towards O in the same x-direction. Hence, A3 = (-4, 0).
Step 4:
We continue this pattern until we find A12. By following the same steps as above, we find:
A4 = (-5, 0)
A5 = (-6, 0)
A6 = (-7, 0)
A7 = (-8, 0)
A8 = (-9, 0)
A9 = (-10, 0)
A10 = (-11, 0)
A11 = (-12, 0)
A12 = (-13, 0)
Therefore, the x-coordinate of A12 is -13.