Two vertices of an equilateral triangle are (0, 0) and (3, √3).

Let the third vertex of the equilaterla triangle be (x, y)

Distance between (0, 0) and (x, y) = Distance between (0, 0) and (3, √3) = Distance between (x, y) and (3, √3)

√(x2 + y2) = √(32 + 3) = √[(x - 3)2 + (y - √3)2]

x2 + y2 = 12

x2 + 9 - 6x + y2 + 3 - 2√3y = 12

24 -  6x - 2√3y = 12

- 6x - 2√3y = - 12

3x + √3y = 6

x = (6 - √3y) / 3

⇒ [(6 - √3y)/3]2 + y2 = 12

⇒ (36 + 3y2 - 12√3y) / 9 + y2 = 12

⇒ 36 + 3y2 - 12√3y + 9y2 = 108

⇒ - 12√3y + 12y2 - 72 = 0

⇒ -√3y + y2 - 6 = 0

⇒ (y - 2√3)(y + √3) = 0

⇒ y = 2√3 or - √3

If y = 2√3, x = (6 - 6) / 3 = 0

If y = -√3, x = (6 + 3) / 3 = 3

So, the third vertex of the equilateral triangle = (0, 2√3) or (3, -√3).



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Given Information:
Two vertices of an equilateral triangle are (0,0) and (3,√3).

Solution:

Step 1: Understand the Problem
The problem asks us to find the coordinates of the third vertex of an equilateral triangle given the coordinates of two vertices.

Step 2: Recall the Properties of an Equilateral Triangle
An equilateral triangle is a special type of triangle where all three sides are equal in length and all three angles are equal to 60 degrees. The distance between any two vertices of an equilateral triangle is equal to the length of the sides.

Step 3: Find the Length of the Sides
To find the length of the sides of the equilateral triangle, we can use the distance formula. The distance between two points (x1, y1) and (x2, y2) is given by the formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)

Using the given vertices, we can calculate the distances as follows:
Side 1: d1 = √((3 - 0)^2 + (√3 - 0)^2) = √(3^2 + (√3)^2) = √(9 + 3) = √12 = 2√3

Step 4: Determine the Third Vertex
Since the equilateral triangle has equal sides, the distance between the two given vertices is equal to the distance between any of the given vertices and the third vertex.

We know that the length of each side of the equilateral triangle is 2√3. Therefore, the distance between the two given vertices and the third vertex is also 2√3.

To find the coordinates of the third vertex, we need to consider the given vertices as the endpoints of a line segment. The midpoint of this line segment will give us the coordinates of the third vertex.

The midpoint coordinates can be found using the midpoint formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

Using the given vertices (0,0) and (3,√3), we can find the midpoint coordinates:
Midpoint = ((0 + 3)/2, (0 + √3)/2) = (3/2, √3/2)

Therefore, the coordinates of the third vertex are (3/2, √3/2).

Step 5: Conclusion
The coordinates of the third vertex of the equilateral triangle, given the vertices (0,0) and (3,√3), are (3/2, √3/2).