Two vertices of `DeltaABC` are `A(-1,4)` and B(5,2) and its centroid is `(0, -3)` . Find the co-ordinates of point C .

Answer

To find the coordinates of point C in triangle ABC, given the vertices A(-1, 4), B(5, 2), and the centroid G(0, -3), we can use the formula for the centroid of a triangle. The centroid G of triangle ABC is given by:

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

Where \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are the coordinates of points A, B, and C respectively.

### Step 1: Assign coordinates
Let:
- \( A(-1, 4) \) → \( (x_1, y_1) = (-1, 4) \)
- \( B(5, 2) \) → \( (x_2, y_2) = (5, 2) \)
- \( C(x, y) \) → \( (x_3, y_3) = (x, y) \)

### Step 2: Set up the centroid equations
The coordinates of the centroid G are given as \( (0, -3) \). Therefore, we can set up the following equations:

1. For the x-coordinate:
\[ \frac{-1 + 5 + x}{3} = 0 \]

2. For the y-coordinate:
\[ \frac{4 + 2 + y}{3} = -3 \]

### Step 3: Solve the x-coordinate equation
From the first equation:
\[ \frac{-1 + 5 + x}{3} = 0 \]
Multiplying both sides by 3:
\[ -1 + 5 + x = 0 \]
Simplifying:
\[ 4 + x = 0 \]
Thus:
\[ x = -4 \]

### Step 4: Solve the y-coordinate equation
From the second equation:
\[ \frac{4 + 2 + y}{3} = -3 \]
Multiplying both sides by 3:
\[ 4 + 2 + y = -9 \]
Simplifying:
\[ 6 + y = -9 \]
Thus:
\[ y = -9 - 6 = -15 \]

### Step 5: Conclusion
The coordinates of point C are:
\[ C(-4, -15) \]

### Final Answer
The coordinates of point C are \((-4, -15)\).

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