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 follow these steps:

### Step 1: Understand the Centroid Formula
The centroid (G) of a triangle with vertices A(x1, y1), B(x2, y2), and C(x3, y3) is given by the formula:
\[ G\left(\frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3}\right) \]
We know the coordinates of the centroid (3, -5) and two vertices A(4, -8) and B(3, 6).

### Step 2: Set Up the Equations
Let the coordinates of the third vertex C be (x, y). According to the centroid formula, we can set up the following equations:
1. For the x-coordinate:
\[ \frac{4 + 3 + x}{3} = 3 \]
2. For the y-coordinate:
\[ \frac{-8 + 6 + y}{3} = -5 \]

### Step 3: Solve for x
From the first equation:
\[ \frac{4 + 3 + x}{3} = 3 \]
Multiply both sides by 3:
\[ 4 + 3 + x = 9 \]
Combine like terms:
\[ 7 + x = 9 \]
Subtract 7 from both sides:
\[ x = 2 \]

### Step 4: Solve for y
From the second equation:
\[ \frac{-8 + 6 + y}{3} = -5 \]
Multiply both sides by 3:
\[ -8 + 6 + y = -15 \]
Combine like terms:
\[ -2 + y = -15 \]
Add 2 to both sides:
\[ y = -13 \]

### Step 5: Write the Coordinates of the Third Vertex
The coordinates of the third vertex C are (x, y) = (2, -13).

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

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