Find the centroid of `Delta ABC` whose vertices are A(2, 2), B(-4, -4) and C(5, -8).

Answer

To find the centroid of triangle ABC with vertices A(2, 2), B(-4, -4), and C(5, -8), we can use the formula for the centroid of a triangle. The centroid (G) can be found using the following formula:

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

Where:
- \( (x_1, y_1) \) are the coordinates of vertex A,
- \( (x_2, y_2) \) are the coordinates of vertex B,
- \( (x_3, y_3) \) are the coordinates of vertex C.

### Step 1: Identify the coordinates of the vertices
- A(2, 2) → \( x_1 = 2, y_1 = 2 \)
- B(-4, -4) → \( x_2 = -4, y_2 = -4 \)
- C(5, -8) → \( x_3 = 5, y_3 = -8 \)

### Step 2: Substitute the coordinates into the centroid formula
\[ G\left(x, y\right) = \left(\frac{2 + (-4) + 5}{3}, \frac{2 + (-4) + (-8)}{3}\right) \]

### Step 3: Calculate the x-coordinate of the centroid
\[ x = \frac{2 - 4 + 5}{3} = \frac{3}{3} = 1 \]

### Step 4: Calculate the y-coordinate of the centroid
\[ y = \frac{2 - 4 - 8}{3} = \frac{-10}{3} \]

### Step 5: Write the coordinates of the centroid
The coordinates of the centroid G are:
\[ G\left(1, -\frac{10}{3}\right) \]

### Final Answer
The centroid of triangle ABC is \( G\left(1, -\frac{10}{3}\right) \).

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