The coordinates of vertices A and B of an equilateral triangle ABC are (- 4, 0) and (4, 0) respectively. Which of the following could be coordinates of C

Options

A. `(0,2,sqrt(3))`
B. (0,4)
C. `(0,4qrt(3))`
D. `(0,3)`

Answer

To find the coordinates of point C in the equilateral triangle ABC where A and B are given as A(-4, 0) and B(4, 0), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Coordinates of A and B**:
- A = (-4, 0)
- B = (4, 0)

2. **Calculate the Length of AB**:
- The distance between points A and B can be calculated using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- Plugging in the coordinates:
\[ AB = \sqrt{(4 - (-4))^2 + (0 - 0)^2} = \sqrt{(4 + 4)^2} = \sqrt{8^2} = 8 \]

3. **Determine the Coordinates of C**:
- Since triangle ABC is equilateral, the length of AC and BC must also be equal to 8.
- We can assume that point C lies on the y-axis, so let C be (0, y).

4. **Set Up the Equation for AC**:
- Using the distance formula for AC:
\[ AC = \sqrt{(0 - (-4))^2 + (y - 0)^2} = \sqrt{(4)^2 + y^2} = \sqrt{16 + y^2} \]
- Set this equal to 8:
\[ \sqrt{16 + y^2} = 8 \]

5. **Square Both Sides**:
- Squaring both sides to eliminate the square root:
\[ 16 + y^2 = 64 \]

6. **Solve for y**:
- Rearranging gives:
\[ y^2 = 64 - 16 = 48 \]
- Taking the square root:
\[ y = \pm \sqrt{48} = \pm 4\sqrt{3} \]

7. **Final Coordinates of C**:
- Therefore, the possible coordinates for point C are:
\[ C(0, 4\sqrt{3}) \quad \text{or} \quad C(0, -4\sqrt{3}) \]

### Conclusion:
The coordinates of point C could be either (0, 4√3) or (0, -4√3).