Two vertices of a triangle are A(1, 1), B (2, -3). If its centroid is (2, 1) find the third vertex.

Answer

To find the third vertex of the triangle given the two vertices A(1, 1) and B(2, -3), and the centroid G(2, 1), 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:
\[ G\left(\frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3}\right) \]
In our case, we know the coordinates of A and B, and we need to find the coordinates of C (let's denote them as (x, y)).

### Step 2: Set Up the Equations
From the centroid formula, we can set up two equations based on the x-coordinates and y-coordinates.

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

For the y-coordinate:
\[ \frac{1 + (-3) + y}{3} = 1 \]

### Step 3: Solve for x
Starting with the x-coordinate equation:
\[ \frac{1 + 2 + x}{3} = 2 \]
Multiply both sides by 3:
\[ 1 + 2 + x = 6 \]
Combine like terms:
\[ 3 + x = 6 \]
Subtract 3 from both sides:
\[ x = 3 \]

### Step 4: Solve for y
Now, let's solve the y-coordinate equation:
\[ \frac{1 - 3 + y}{3} = 1 \]
Multiply both sides by 3:
\[ 1 - 3 + y = 3 \]
Combine like terms:
\[ -2 + y = 3 \]
Add 2 to both sides:
\[ y = 5 \]

### Step 5: Conclusion
The coordinates of the third vertex C are (3, 5).

Thus, the third vertex of the triangle is C(3, 5).

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