To find the centroid of the triangle with one vertex at (1, 1) and midpoints of two sides through this vertex at (-1, 2) and (3, 2), we can follow these steps:
### Step 1: Identify the given points
- Vertex \( A = (1, 1) \)
- Midpoint \( M_1 = (-1, 2) \)
- Midpoint \( M_2 = (3, 2) \)
### Step 2: Find the coordinates of the other two vertices
Let the other two vertices be \( B \) and \( C \).
#### For vertex \( B \):
The midpoint \( M_1 \) is given by the formula:
\[
M_1 = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Substituting the known values:
\[
(-1, 2) = \left( \frac{1 + x_2}{2}, \frac{1 + y_2}{2} \right)
\]
From the x-coordinate:
\[
-1 = \frac{1 + x_2}{2} \implies -2 = 1 + x_2 \implies x_2 = -3
\]
From the y-coordinate:
\[
2 = \frac{1 + y_2}{2} \implies 4 = 1 + y_2 \implies y_2 = 3
\]
Thus, vertex \( B = (-3, 3) \).
#### For vertex \( C \):
The midpoint \( M_2 \) is given by:
\[
M_2 = \left( \frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2} \right)
\]
Substituting the known values:
\[
(3, 2) = \left( \frac{1 + x_3}{2}, \frac{1 + y_3}{2} \right)
\]
From the x-coordinate:
\[
3 = \frac{1 + x_3}{2} \implies 6 = 1 + x_3 \implies x_3 = 5
\]
From the y-coordinate:
\[
2 = \frac{1 + y_3}{2} \implies 4 = 1 + y_3 \implies y_3 = 3
\]
Thus, vertex \( C = (5, 3) \).
### Step 3: Calculate the centroid of the triangle
The formula for the centroid \( G \) of a triangle with vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) 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 of the vertices:
\[
G = \left( \frac{1 + (-3) + 5}{3}, \frac{1 + 3 + 3}{3} \right)
\]
Calculating the x-coordinate:
\[
G_x = \frac{1 - 3 + 5}{3} = \frac{3}{3} = 1
\]
Calculating the y-coordinate:
\[
G_y = \frac{1 + 3 + 3}{3} = \frac{7}{3}
\]
Thus, the centroid \( G = \left( 1, \frac{7}{3} \right) \).
### Final Answer
The centroid of the triangle is \( \left( 1, \frac{7}{3} \right) \).
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