The centroid of the triangle formed by (2, -5), (2, 7), (4, 7) is

Answer

To find the centroid of the triangle formed by the points (2, -5), (2, 7), and (4, 7), we can use the formula for the centroid of a triangle given by the coordinates of its vertices. The formula for the centroid \( C(x, y) \) is:

\[ C\left(x, y\right) = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]

Where \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are the coordinates of the vertices of the triangle.

### Step 1: Identify the coordinates of the vertices.
The vertices of the triangle are:
- \( (x_1, y_1) = (2, -5) \)
- \( (x_2, y_2) = (2, 7) \)
- \( (x_3, y_3) = (4, 7) \)

### Step 2: Plug the coordinates into the centroid formula.
Now we substitute the values into the centroid formula:

\[ C\left(x, y\right) = \left(\frac{2 + 2 + 4}{3}, \frac{-5 + 7 + 7}{3}\right) \]

### Step 3: Calculate the x-coordinate of the centroid.
Calculating the x-coordinate:

\[ x = \frac{2 + 2 + 4}{3} = \frac{8}{3} \]

### Step 4: Calculate the y-coordinate of the centroid.
Calculating the y-coordinate:

\[ y = \frac{-5 + 7 + 7}{3} = \frac{9}{3} = 3 \]

### Step 5: Write the final coordinates of the centroid.
Thus, the coordinates of the centroid \( C \) are:

\[ C\left(\frac{8}{3}, 3\right) \]

### Final Answer:
The centroid of the triangle formed by the points (2, -5), (2, 7), and (4, 7) is \( \left(\frac{8}{3}, 3\right) \).

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