If a vertex of a triangle is (1, 1) and the mid-points of two side through this vertex are (-1, 2) and (3, 2), then centroid of the triangle is

Options

A. `((1)/(3), (7)/(3))`
B. `(1,(7)/(3))`
C. `(-(1)/(3),(7)/(3))`
D. `(-1,(7)/(3))`

Answer

To find the centroid of the triangle given one vertex and the midpoints of the other two sides, we can follow these steps:

### Step 1: Identify the given points
- The vertex of the triangle \( A \) is given as \( (1, 1) \).
- The midpoints of the sides through this vertex are \( M_1 = (-1, 2) \) and \( M_2 = (3, 2) \).

### Step 2: Use the midpoint formula to find the other vertices
The midpoint formula states that if \( M \) is the midpoint of segment \( AB \), then:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
From this, we can derive the coordinates of the vertices \( B \) and \( C \).

#### Finding vertex \( B \):
Let the coordinates of vertex \( B \) be \( (x_2, y_2) \).
Using the midpoint \( M_1 = (-1, 2) \):
\[ \frac{x_2 + 1}{2} = -1 \quad \text{and} \quad \frac{y_2 + 1}{2} = 2 \]
Solving these equations:
1. For \( x_2 \):
\[ x_2 + 1 = -2 \implies x_2 = -3 \]
2. For \( y_2 \):
\[ y_2 + 1 = 4 \implies y_2 = 3 \]
Thus, the coordinates of vertex \( B \) are \( (-3, 3) \).

#### Finding vertex \( C \):
Let the coordinates of vertex \( C \) be \( (x_3, y_3) \).
Using the midpoint \( M_2 = (3, 2) \):
\[ \frac{x_3 + 1}{2} = 3 \quad \text{and} \quad \frac{y_3 + 1}{2} = 2 \]
Solving these equations:
1. For \( x_3 \):
\[ x_3 + 1 = 6 \implies x_3 = 5 \]
2. For \( y_3 \):
\[ y_3 + 1 = 4 \implies y_3 = 3 \]
Thus, the coordinates of vertex \( C \) are \( (5, 3) \).

### Step 3: Calculate the centroid of the triangle
The centroid \( G \) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) is given by:
\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
Substituting the coordinates:
- \( A(1, 1) \)
- \( B(-3, 3) \)
- \( C(5, 3) \)

Calculating the x-coordinate of the centroid:
\[ G_x = \frac{1 + (-3) + 5}{3} = \frac{3}{3} = 1 \]

Calculating the y-coordinate of the centroid:
\[ G_y = \frac{1 + 3 + 3}{3} = \frac{7}{3} \]

Thus, the coordinates of the centroid \( G \) are:
\[ G = \left( 1, \frac{7}{3} \right) \]

### Final Answer:
The centroid of the triangle is \( \left( 1, \frac{7}{3} \right) \).

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