Two vertices of a triangle are (1,3) and (4,-5) . If its centroid is (7,2) , then the third vertex is

Options

A. (16,8)
B. (8,12)
C. (12,8)
D. (8,16)

Answer

To find the third vertex of the triangle given two vertices and the centroid, we can use the formula for the centroid of a triangle.

### Step-by-Step Solution:

1. **Identify the Given Points:**
- Let the vertices of the triangle be \( A(1, 3) \) and \( B(4, -5) \).
- The centroid \( G \) is given as \( (7, 2) \).
- Let the third vertex be \( C(x_3, y_3) \).

2. **Use the Centroid Formula:**
The coordinates of the centroid \( G \) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) are given by:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]

3. **Set Up the Equations:**
From the centroid coordinates, we can set up two equations:
- For the x-coordinate:
\[ 7 = \frac{1 + 4 + x_3}{3} \]
- For the y-coordinate:
\[ 2 = \frac{3 + (-5) + y_3}{3} \]

4. **Solve for \( x_3 \):**
Multiply both sides of the x-coordinate equation by 3:
\[ 21 = 1 + 4 + x_3 \]
Simplifying gives:
\[ 21 = 5 + x_3 \implies x_3 = 21 - 5 = 16 \]

5. **Solve for \( y_3 \):**
Multiply both sides of the y-coordinate equation by 3:
\[ 6 = 3 - 5 + y_3 \]
Simplifying gives:
\[ 6 = -2 + y_3 \implies y_3 = 6 + 2 = 8 \]

6. **Conclusion:**
Thus, the coordinates of the third vertex \( C \) are \( (16, 8) \).

### Final Answer:
The third vertex is \( C(16, 8) \).