The length of the side of an equilateral triangle inscribed in the parabola, `y^2=4x` so that one of its angular point is at the vertex is:

Answer

To find the length of the side of an equilateral triangle inscribed in the parabola \( y^2 = 4x \) with one vertex at the origin (the vertex of the parabola), we can follow these steps:

### Step 1: Understand the Parabola
The given parabola is \( y^2 = 4x \). The vertex of this parabola is at the point \( (0, 0) \).

**Hint:** The vertex of the parabola is the point where the parabola changes direction.

### Step 2: Set Up the Triangle
Let the vertices of the equilateral triangle be \( A(0, 0) \) (the vertex at the origin), and the other two vertices \( B \) and \( C \) will lie on the parabola.

### Step 3: Use Polar Coordinates
We can express the coordinates of points \( B \) and \( C \) using polar coordinates. Let the angle \( \theta \) be the angle that the line from the origin to point \( B \) makes with the x-axis. The coordinates of point \( B \) can be expressed as:
- \( B = (r \cos \theta, r \sin \theta) \)

### Step 4: Determine the Angles
Since the triangle is equilateral, the angle at vertex \( A \) is \( 60^\circ \). Therefore, the angles at points \( B \) and \( C \) will be \( 30^\circ \) each (as the triangle is symmetric about the x-axis).

### Step 5: Calculate Coordinates of Points B and C
Using the angles:
- For point \( B \):
- \( x_B = r \cos(30^\circ) = r \cdot \frac{\sqrt{3}}{2} \)
- \( y_B = r \sin(30^\circ) = r \cdot \frac{1}{2} \)

Thus, the coordinates of point \( B \) are:
- \( B = \left( r \cdot \frac{\sqrt{3}}{2}, r \cdot \frac{1}{2} \right) \)

### Step 6: Substitute into the Parabola Equation
Since point \( B \) lies on the parabola, we substitute \( x_B \) and \( y_B \) into the parabola equation \( y^2 = 4x \):
\[ \left( r \cdot \frac{1}{2} \right)^2 = 4 \left( r \cdot \frac{\sqrt{3}}{2} \right) \]
This simplifies to:
\[ \frac{r^2}{4} = 4 \cdot \frac{r \sqrt{3}}{2} \]
\[ \frac{r^2}{4} = 2r \sqrt{3} \]

### Step 7: Solve for r
Multiplying both sides by 4 to eliminate the fraction:
\[ r^2 = 8r \sqrt{3} \]
Rearranging gives:
\[ r^2 - 8r \sqrt{3} = 0 \]
Factoring out \( r \):
\[ r(r - 8\sqrt{3}) = 0 \]
Thus, \( r = 0 \) or \( r = 8\sqrt{3} \). Since \( r = 0 \) does not provide a valid triangle, we take:
\[ r = 8\sqrt{3} \]

### Step 8: Find the Length of the Side of the Triangle
The length of each side of the equilateral triangle is \( r \). Therefore, the length of the side of the triangle is:
\[ \text{Length of side} = 8\sqrt{3} \]

### Final Answer
The length of the side of the equilateral triangle inscribed in the parabola \( y^2 = 4x \) with one vertex at the origin is \( 8\sqrt{3} \).

---