The centroid of an equilateral triangle is (0, 0) and the length of the altitude is 6. The equation of the circumcirele of the triangle is

Options

A. `x^(2)+y^(2)=6`
B. `x^(2)+y^(2)=16`
C. `x^(2)+y^(2)=9`
D. `x^(2)+y^(2)=36`

Answer

To find the equation of the circumcircle of an equilateral triangle with a centroid at (0, 0) and an altitude of 6, we can follow these steps:

### Step 1: Understand the properties of the equilateral triangle.
The centroid of an equilateral triangle divides each median in a 2:1 ratio. The length of the altitude is equal to the length of the median.

### Step 2: Calculate the length of the median.
Given that the altitude (which is also the median) is 6, we can denote the length of the median \(AD\) as:
\[ AD = 6 \]

### Step 3: Find the length from the vertex to the centroid.
Using the property of the centroid dividing the median in a 2:1 ratio, we can find the length from the vertex \(A\) to the centroid \(G\):
\[ AG = \frac{2}{3} \times AD = \frac{2}{3} \times 6 = 4 \]

### Step 4: Determine the circumradius.
In an equilateral triangle, the circumradius \(R\) is equal to the length from the centroid to any vertex. Thus:
\[ R = AG = 4 \]

### Step 5: Write the equation of the circumcircle.
The general equation of a circle with center \((h, k)\) and radius \(r\) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Here, the center \(G\) of the circumcircle is at \((0, 0)\) and the radius \(R\) is 4. Substituting these values into the equation gives:
\[ (x - 0)^2 + (y - 0)^2 = 4^2 \]
This simplifies to:
\[ x^2 + y^2 = 16 \]

### Final Answer:
The equation of the circumcircle of the triangle is:
\[ x^2 + y^2 = 16 \]

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