Find the coordinates of third vertex of a triangle, if centroid of the triangle is (3, - 5) and two of its vertices are (4, -8) and (3, 6).

Options

A. (1, 5)
B. (2, -13)
C. (5, 6)
D. (-1, 3)

Answer

To find the coordinates of the third vertex of a 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 A(x1, y1), B(x2, y2), and C(x3, y3) is given by:

\[ G = \left( \frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3} \right) \]

Given:
- Centroid \( G(3, -5) \)
- Vertex A \( (4, -8) \)
- Vertex B \( (3, 6) \)

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

### Step 1: Set up the equations for the centroid
From the formula for the centroid, we can set up the following equations based on the x-coordinates and y-coordinates:

1. For the x-coordinates:
\[ \frac{4 + 3 + x}{3} = 3 \]

2. For the y-coordinates:
\[ \frac{-8 + 6 + y}{3} = -5 \]

### Step 2: Solve for x
Multiply the first equation by 3 to eliminate the fraction:
\[ 4 + 3 + x = 9 \]
Combine like terms:
\[ 7 + x = 9 \]
Now, solve for \( x \):
\[ x = 9 - 7 = 2 \]

### Step 3: Solve for y
Multiply the second equation by 3 to eliminate the fraction:
\[ -8 + 6 + y = -15 \]
Combine like terms:
\[ -2 + y = -15 \]
Now, solve for \( y \):
\[ y = -15 + 2 = -13 \]

### Conclusion
The coordinates of the third vertex C are \( (2, -13) \).

### Final Answer
The coordinates of the third vertex are \( (2, -13) \).

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