How many equilateral triangles in the plane have two vertices in the set {(0,0), (0,1),(1,0),(1,1)}?
I'd say 12
two on each of 4 sides,
two on each of 2 diagonals.
To find the number of equilateral triangles with two vertices in the set {(0,0), (0,1),(1,0),(1,1)}, we will go through the steps as follows:
Step 1: Consider one of the vertices as (0,0).
Step 2: Determine the possible positions for the other vertex with respect to (0,0) such that they form an equilateral triangle. Since an equilateral triangle has three equal sides, we need to find the distances between the two vertices. The possible distances are:
a) √2, which correspond to the vertices (0,1) and (1,0).
b) √2 * √3 = √6, which corresponds to the vertex (1,1).
Step 3: Consider the other vertex as (0,1).
Step 4: Determine the possible positions for the third vertex with respect to (0,1) such that they form an equilateral triangle. The possible distances are the same as in Step 2.
Step 5: Consider the other vertex as (1,0).
Step 6: Determine the possible positions for the third vertex with respect to (1,0) such that they form an equilateral triangle. The possible distances are the same as in Step 2.
Step 7: Consider the other vertex as (1,1).
Step 8: Determine the possible positions for the third vertex with respect to (1,1) such that they form an equilateral triangle. The possible distances are the same as in Step 2.
Step 9: Add up the number of triangles formed in each case.
The total number of equilateral triangles with two vertices in the set {(0,0), (0,1),(1,0),(1,1)} is the sum of the triangles formed in each of the above steps.
To determine the number of equilateral triangles in the plane with two vertices in the set {(0,0), (0,1), (1,0), (1,1)}, we can approach it by considering all possible pairs of points from the given set and checking if a third point exists such that it forms an equilateral triangle.
Let's break down the solution step by step:
Step 1: Consider all possible pairs of points from the given set. There are four points in total, so we need to choose two of them for each pair. Using combinatorics, we have a total of C(4, 2) pairs, which is equal to 6 pairs.
The six possible pairs are:
{(0,0), (0,1)}, {(0,0), (1,0)}, {(0,0), (1,1)}
{(0,1), (1,0)}, {(0,1), (1,1)}, {(1,0), (1,1)}
Step 2: For each pair, check if there exists a third point such that it forms an equilateral triangle with the given pair.
To determine the third point, we can examine the distances between the two chosen points for each pair. If the two points are (x1, y1) and (x2, y2), the distance between them is calculated using the Euclidean distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
For an equilateral triangle, all three sides must have equal lengths. In this case, we need to check if the distance between the two chosen points is equal to the distance between either of the chosen points and a third point.
Step 3: Check if a third point exists for each possible pair.
For every pair, we calculate the distances between its two points, and then we check if either point A or point B has the same distance to a third point C. If such a point C exists, it means that an equilateral triangle can be formed.
Let's go through each pair and check for the existence of a third point.
1. {(0,0), (0,1)}
Distance between (0,0) and (0,1) = sqrt((0 - 0)^2 + (1 - 0)^2) = 1
The possible third point C could lie at the same distance from either of the chosen points, i.e., the distance from C to (0,0) or C to (0,1) should be 1.
After checking all points in the given set, we find no point that satisfies the condition. Hence, there are no equilateral triangles with two vertices from this pair.
2. {(0,0), (1,0)}
Distance between (0,0) and (1,0) = sqrt((1 - 0)^2 + (0 - 0)^2) = 1
Here, the possible third point C could be (0,-1) or (2,-1) since they are at a distance of 1 from (0,0) and (1,0), respectively.
Therefore, there is one equilateral triangle with two vertices from this pair:
{(0,0), (1,0), (0,-1)} or {(0,0), (1,0), (2,-1)}
3. {(0,0), (1,1)}
Distance between (0,0) and (1,1) = sqrt((1 - 0)^2 + (1 - 0)^2) = sqrt(2)
Here, no point in the given set satisfies the condition of being at a distance of sqrt(2) from both (0,0) and (1,1).
Hence, there are no equilateral triangles with two vertices from this pair.
4. {(0,1), (1,0)}
Distance between (0,1) and (1,0) = sqrt((1 - 0)^2 + (0 - 1)^2) = sqrt(2)
No point in the given set satisfies the condition of being at a distance of sqrt(2) from both (0,1) and (1,0).
Therefore, there are no equilateral triangles with two vertices from this pair.
5. {(0,1), (1,1)}
Distance between (0,1) and (1,1) = sqrt((1 - 0)^2 + (1 - 1)^2) = 1
Here, the possible third point C could be (1,2) or (-1,2) since they are at a distance of 1 from (0,1) and (1,1), respectively.
Thus, there is one equilateral triangle with two vertices from this pair:
{(0,1), (1,1), (1,2)} or {(0,1), (1,1), (-1,2)}
6. {(1,0), (1,1)}
Distance between (1,0) and (1,1) = sqrt((1 - 1)^2 + (1 - 0)^2) = 1
Here, no point in the given set satisfies the condition of being at a distance of 1 from both (1,0) and (1,1).
Hence, there are no equilateral triangles with two vertices from this pair.
In summary, there are 2 equilateral triangles in the plane with two vertices in the set {(0,0), (0,1), (1,0), (1,1)}:
{(0,0), (1,0), (0,-1)} or {(0,0), (1,0), (2,-1)}
{(0,1), (1,1), (1,2)} or {(0,1), (1,1), (-1,2)}
Therefore, the answer is 2.