An equilateral triangle, ABC, has two of its vertices, A and B, below the x-axis; has the third vertex C above the x-axis, and contains the points (0,0) on segment AC, and (0,1) on segment BC. How long is the path traced out by all possible points C to two decimal places?
Do we assume that there is a single triangle, of fixed size, which we are manipulating? That is, its sides are of fixed length?
Assume the triangle is of fixed size. The sides are a fixed length.
To find the length of the path traced out by all possible points C, we need to determine the range of values for the x-coordinate of vertex C.
Let's consider the equilateral triangle ABC. Since two of its vertices, A and B, are below the x-axis, we can assume they have negative y-coordinates. We know that point A lies on the x-axis, so its y-coordinate is 0. Point B lies on segment BC and has coordinates (0,1), so we can deduce that the y-coordinate of B is 1.
Since triangle ABC is equilateral, the distance between points A and B is equal to the distance between points A and C, as well as the distance between points B and C. As the y-coordinate of point A is 0 and the y-coordinate of point B is 1, the height of the equilateral triangle is 1. Since the triangle is equilateral, the length of side AB is also equal to 1.
Let's consider the x-coordinate of vertex C. Vertex C lies above the x-axis, meaning its y-coordinate is positive. To determine the range of x-coordinates for vertex C, we need to calculate the distance between points A and C.
The length of side AC is equal to the x-coordinate of vertex C. The length of side AB is equal to 1. The length of side BC is equal to the distance between points A and B, which is also equal to 1. Since triangle ABC is equilateral, all sides are equal in length.
Using the properties of an equilateral triangle, we can calculate the length of side AC:
AC = √(AB^2 - BC^2) = √(1^2 - 1^2) = √0 = 0
Therefore, the x-coordinate of vertex C is 0. This means that all possible points C lie on the y-axis.
The length of the path traced out by all possible points C is the range of values for the y-coordinate of vertex C, which is the same as the range of y-coordinates for points A and B. In this case, the range is from 0 (y-coordinate of A) to 1 (y-coordinate of B).
Hence, the length of the path traced out by all possible points C is 1 unit.