The question in the image is in Persian, and it translates to the following:

Question: If two vertices of an equilateral triangle are $(0, 3)$ and $(0, -3)$, where is the third vertex? Show this geometrically by drawing the triangle and explain why two distinct solutions exist.

Answer:

This problem asks for the location of the third vertex of an equilateral triangle, where the given two vertices are $(0, 3)$ and $(0, -3)$. Since the triangle is equilateral, the distances between all three vertices must be equal.

To solve this:

1. We know the length of the side between the points $(0, 3)$ and $(0, -3)$, which is the vertical distance: $d = 3 - (-3) = 6$

2. The distance between any two vertices of an equilateral triangle must be equal to this 6-unit distance. To find the third vertex, we must look for a point that is 6 units away from both $(0, 3)$ and $(0, -3)$.

3. The third vertex will lie on a circle centered at $(0, 0)$ with a radius of 6 units, but it must also satisfy the geometric condition of forming an equilateral triangle with the two given points. This results in two possible positions for the third vertex (since rotating the triangle clockwise or counterclockwise by 60° will yield two different valid triangles).

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