Centroid Formula Basics

The centroid of a triangle is the point where its three medians intersect. A median connects a vertex to the midpoint of the opposite side. If a triangle has vertices with coordinates $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, the coordinates of its centroid $(G_x, G_y)$ are calculated using the following formula:

$$ G_x = \frac{x_1 + x_2 + x_3}{3} $$

$$ G_y = \frac{y_1 + y_2 + y_3}{3} $$

Triangle Vertex and Centroid Details

We are given the coordinates of the centroid of a triangle and the coordinates of its three vertices. One of the vertex coordinates includes an unknown value, p, which we need to find.

• Centroid coordinates $(G_x, G_y)$ = (3, 2)
• Vertex 1 coordinates $(x_1, y_1)$ = (0, 4)
• Vertex 2 coordinates $(x_2, y_2)$ = (-2, 0)
• Vertex 3 coordinates $(x_3, y_3)$ = (p, 2)

Calculating Vertex 'p' Coordinate

We can use the centroid formula and the given coordinates to set up equations and solve for p. We will apply the formula separately for the x-coordinate and the y-coordinate.

Using the x-coordinate Formula

Substitute the known x-coordinates into the formula for the centroid's x-coordinate:

$$ G_x = \frac{x_1 + x_2 + x_3}{3} $$

$$ 3 = \frac{0 + (-2) + p}{3} $$

Now, we solve this equation for p:

$$ 3 = \frac{-2 + p}{3} $$

Multiply both sides by 3:

$$ 3 \times 3 = -2 + p $$

$$ 9 = -2 + p $$

Add 2 to both sides to isolate p:

$$ 9 + 2 = p $$

$$ 11 = p $$

Verifying with the y-coordinate Formula

Let's check if the given y-coordinates are consistent with the centroid's y-coordinate:

$$ G_y = \frac{y_1 + y_2 + y_3}{3} $$

$$ 2 = \frac{4 + 0 + 2}{3} $$

$$ 2 = \frac{6}{3} $$

$$ 2 = 2 $$

The y-coordinates match, confirming our setup.

Final Result for Vertex 'p'

Based on the calculations using the centroid formula for the x-coordinates, the value of p is 11.