The centroid of a triangle is (2,3) and two of its vertices are (5,6) and (-1,4). The thrid vertex of the triangle is

Options

A. (2,1)
B. (2,-1)
C. (1,2)
D. (1,-2)

Answer

To find the third vertex of the triangle given the centroid and two vertices, we can use the formula for the centroid of a triangle. The centroid (G) of a triangle with vertices at points A(x1, y1), B(x2, y2), and C(x3, y3) is given by:

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

Given:
- Centroid \( G(2, 3) \)
- Vertex \( B(5, 6) \)
- Vertex \( C(-1, 4) \)

Let the coordinates of the third vertex \( A \) be \( (x, y) \).

### Step 1: Set up the equations for the centroid
Using the formula for the centroid, we can set up the following equations:

1. For the x-coordinates:
\[ \frac{x + 5 - 1}{3} = 2 \]

2. For the y-coordinates:
\[ \frac{y + 6 + 4}{3} = 3 \]

### Step 2: Solve the first equation for x
Multiply both sides of the first equation by 3:
\[ x + 5 - 1 = 6 \]
Simplify:
\[ x + 4 = 6 \]
Subtract 4 from both sides:
\[ x = 2 \]

### Step 3: Solve the second equation for y
Multiply both sides of the second equation by 3:
\[ y + 6 + 4 = 9 \]
Simplify:
\[ y + 10 = 9 \]
Subtract 10 from both sides:
\[ y = -1 \]

### Step 4: Write the coordinates of the third vertex
The coordinates of the third vertex \( A \) are \( (x, y) = (2, -1) \).

### Final Answer
Thus, the third vertex of the triangle is \( (2, -1) \).

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