If two vertices of a triangle are (5, 4) and (-2, 4) and centroid is (5, 6), then third vertex is

Options

A. (12, 10)
B. (10, 12)
C. (-10, 12)
D. (12, -10)

Answer

To find the third vertex of the triangle given two vertices and the centroid, we can follow these steps:

### Step 1: Identify the given points
We have two vertices of the triangle:
- Vertex A (x1, y1) = (5, 4)
- Vertex B (x2, y2) = (-2, 4)

The coordinates of the centroid (G) are given as:
- Centroid G (xg, yg) = (5, 6)

### Step 2: Use the centroid formula
The formula for the centroid of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is:
\[ x_g = \frac{x_1 + x_2 + x_3}{3} \]
\[ y_g = \frac{y_1 + y_2 + y_3}{3} \]

### Step 3: Set up the equations
Let the coordinates of the third vertex be (x3, y3) = (a, b).

Using the centroid formulas, we can set up the following equations:
1. For the x-coordinate:
\[ 5 = \frac{5 + (-2) + a}{3} \]
2. For the y-coordinate:
\[ 6 = \frac{4 + 4 + b}{3} \]

### Step 4: Solve for 'a' (the x-coordinate of the third vertex)
From the first equation:
\[ 5 = \frac{5 - 2 + a}{3} \]
\[ 5 = \frac{3 + a}{3} \]
Multiplying both sides by 3:
\[ 15 = 3 + a \]
Subtracting 3 from both sides:
\[ a = 12 \]

### Step 5: Solve for 'b' (the y-coordinate of the third vertex)
From the second equation:
\[ 6 = \frac{4 + 4 + b}{3} \]
\[ 6 = \frac{8 + b}{3} \]
Multiplying both sides by 3:
\[ 18 = 8 + b \]
Subtracting 8 from both sides:
\[ b = 10 \]

### Step 6: State the third vertex
Thus, the coordinates of the third vertex are:
\[ \text{Third vertex} = (a, b) = (12, 10) \]

### Final Answer
The third vertex of the triangle is (12, 10).

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