Two vertices of an equilateral triangle are origin and (4, 0). What is the area of the triangle ?

Options

A. A sq units
B. V3 sq units
C. `4sqrt(3)` sq units
D. 2V3 sq units

Answer

To find the area of the equilateral triangle with vertices at the origin (0, 0) and (4, 0), we can follow these steps:

### Step 1: Identify the vertices
The vertices of the triangle are:
- A (0, 0) - the origin
- B (4, 0) - the point on the x-axis
- C - the third vertex, which we need to determine.

### Step 2: Calculate the length of side AB
The distance between points A and B can be calculated using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of A and B:
\[ AB = \sqrt{(4 - 0)^2 + (0 - 0)^2} = \sqrt{4^2} = \sqrt{16} = 4 \]

### Step 3: Use the properties of an equilateral triangle
In an equilateral triangle, all sides are equal. Therefore, the length of each side (a) is 4.

### Step 4: Calculate the area of the equilateral triangle
The area \(A\) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Substituting \(a = 4\):
\[ A = \frac{\sqrt{3}}{4} \times 4^2 = \frac{\sqrt{3}}{4} \times 16 = 4\sqrt{3} \]

### Final Answer
The area of the triangle is \(4\sqrt{3}\).

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