The centroid of the triangle whose vertices are (4, - 8), (- 9,7) and (8,13) is _______

Answer

To find the centroid of a triangle given its vertices, we can use the formula for the centroid (G) of a triangle with vertices at \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \). The formula is:

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

Given the vertices of the triangle:
- \( A(4, -8) \) where \( x_1 = 4 \) and \( y_1 = -8 \)
- \( B(-9, 7) \) where \( x_2 = -9 \) and \( y_2 = 7 \)
- \( C(8, 13) \) where \( x_3 = 8 \) and \( y_3 = 13 \)

Now, we will calculate the coordinates of the centroid step by step.

### Step 1: Calculate the x-coordinate of the centroid
Using the formula for the x-coordinate:

\[ x_G = \frac{x_1 + x_2 + x_3}{3} \]

Substituting the values:

\[ x_G = \frac{4 + (-9) + 8}{3} \]

Calculating the sum:

\[ x_G = \frac{4 - 9 + 8}{3} = \frac{3}{3} = 1 \]

### Step 2: Calculate the y-coordinate of the centroid
Using the formula for the y-coordinate:

\[ y_G = \frac{y_1 + y_2 + y_3}{3} \]

Substituting the values:

\[ y_G = \frac{-8 + 7 + 13}{3} \]

Calculating the sum:

\[ y_G = \frac{-8 + 7 + 13}{3} = \frac{12}{3} = 4 \]

### Final Step: Write the coordinates of the centroid
Combining the x and y coordinates, we find the centroid \( G \):

\[ G(1, 4) \]

Thus, the centroid of the triangle whose vertices are \( (4, -8) \), \( (-9, 7) \), and \( (8, 13) \) is \( (1, 4) \).

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