The two vertices of a triangle are (6, 3) and (–1, 7) and its centroid is (1, 5). Find the third vertex.
(a) -2,5
(b)2,5
(c)2,4
(d)3,6

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. The centroid (G) of a triangle with vertices 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:
- Vertex A (x1, y1) = (6, 3)
- Vertex B (x2, y2) = (-1, 7)
- Centroid G = (1, 5)

Let the third vertex C be (x3, y3).

### Step 1: Set up the equations for the x-coordinate of the centroid.
Using the formula for the x-coordinate of the centroid:

\[ 1 = \frac{6 + (-1) + x_3}{3} \]

### Step 2: Simplify the equation.
Multiply both sides by 3 to eliminate the fraction:

\[ 3 = 6 - 1 + x_3 \]

This simplifies to:

\[ 3 = 5 + x_3 \]

### Step 3: Solve for x3.
Subtract 5 from both sides:

\[ x_3 = 3 - 5 = -2 \]

### Step 4: Set up the equations for the y-coordinate of the centroid.
Using the formula for the y-coordinate of the centroid:

\[ 5 = \frac{3 + 7 + y_3}{3} \]

### Step 5: Simplify the equation.
Multiply both sides by 3:

\[ 15 = 3 + 7 + y_3 \]

This simplifies to:

\[ 15 = 10 + y_3 \]

### Step 6: Solve for y3.
Subtract 10 from both sides:

\[ y_3 = 15 - 10 = 5 \]

### Conclusion:
The third vertex C is (-2, 5).

Thus, the answer is (a) -2, 5.

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