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 construction of the points Ai.
Given that A0 is (-1, 0) and O is the origin (0, 0), we will construct each point Ai according to the given conditions.
1. First, let's find the length of |Ai−1Ai| and |Ai−1O| for any i.
Since |Ai−1Ai| is equal to |Ai−1O|, it means that the length of the line segment Ai−1Ai is the same as the distance from the point Ai−1 to the origin O.
2. Now, let's construct A1 based on the given conditions.
Since |A0O| is a distance along the x-axis, we construct a line segment of length |A0O| = 1 along the positive x-axis, starting from A0.
So, A1 will be at the point (2, 0) since we moved 1 unit along the positive x-axis from A0.
3. Next, we construct A2 based on the given conditions.
To construct A2, we need to find the length of |A1O|, which is the same as the distance from A1 to the origin O.
Since |A1O| = |A0O| = 1, we construct a line segment of length |A1O| = 1 perpendicular to the positive x-axis, starting from A1. This line segment will be along the positive y-axis.
So, A2 will be at the point (2, 1) since we moved 1 unit along the positive y-axis from A1.
4. Now, let's continue constructing the points Ai for i ≥ 3 based on the same pattern.
For any i ≥ 3, we need to find the length of |Ai−1O|, which is the same as the distance from Ai−1 to the origin O.
Since |Ai−1O| = |Ai−2O| = 1, we construct a line segment of length |Ai−1O| = 1 perpendicular to the positive y-axis, starting from Ai−1. This line segment will be along the negative x-axis for even values of i and the positive x-axis for odd values of i.
We will alternate between moving a distance of 1 unit along the negative x-axis and 1 unit along the positive x-axis as we construct each point Ai.
5. Repeat steps 3 and 4 for each subsequent point to construct A12.
We continue to construct each point Ai by finding the length of |Ai−1−O| and constructing a line segment along the x or y-axis based on the pattern described in step 4.
To find A12, we need to perform the steps above starting from A0 and constructing points A1, A2, A3, ..., A12.
By following the steps, we will reach the point A12, and the x-coordinate will be the desired answer.