Two vertices of a triangle are (-1, 4) and (5, 2). If its centroid is (0, -3), find the third vertex.

Answer

To find the third vertex of the triangle given two vertices and the centroid, we can follow these steps:

### Step 1: Identify the given points
Let the vertices of the triangle be:
- Vertex A: \( A(-1, 4) \)
- Vertex B: \( B(5, 2) \)
- Vertex C: \( C(h, k) \) (where \( h \) and \( k \) are the coordinates of the third vertex)
- Centroid G: \( G(0, -3) \)

### Step 2: Use the centroid formula
The formula for the centroid \( G \) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) is given by:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]
Substituting the known values:
\[ G\left(\frac{-1 + 5 + h}{3}, \frac{4 + 2 + k}{3}\right) = (0, -3) \]

### Step 3: Set up equations from the centroid
From the x-coordinate of the centroid:
\[ \frac{-1 + 5 + h}{3} = 0 \]
Multiplying both sides by 3:
\[ -1 + 5 + h = 0 \]
Simplifying:
\[ 4 + h = 0 \implies h = -4 \]

From the y-coordinate of the centroid:
\[ \frac{4 + 2 + k}{3} = -3 \]
Multiplying both sides by 3:
\[ 4 + 2 + k = -9 \]
Simplifying:
\[ 6 + k = -9 \implies k = -9 - 6 = -15 \]

### Step 4: Write the coordinates of the third vertex
Thus, the coordinates of the third vertex \( C \) are:
\[ C(-4, -15) \]

### Final Answer
The third vertex of the triangle is \( C(-4, -15) \).

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