If a vertex of a triangle be (1,1) and the middle points of the two sides through it be (-2,3) and (5,2) then the centroid of the triangle is

Options

A. `(5/3,3)`
B. `(5/3,-3)`
C. `(-5/3,3)`
D. `(-5/3,-3)`

Answer

To find the centroid of the triangle with one vertex at \( A(1, 1) \) and midpoints of the other two sides at \( D(-2, 3) \) and \( E(5, 2) \), we will follow these steps:

### Step 1: Determine the coordinates of the other two vertices (B and C) using the midpoint formula.

The midpoint \( D \) of segment \( AB \) is given by:
\[ D = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) \]
Substituting the coordinates:
\[ (-2, 3) = \left( \frac{1 + x_B}{2}, \frac{1 + y_B}{2} \right) \]
From the x-coordinates:
\[ -2 = \frac{1 + x_B}{2} \implies -4 = 1 + x_B \implies x_B = -5 \]
From the y-coordinates:
\[ 3 = \frac{1 + y_B}{2} \implies 6 = 1 + y_B \implies y_B = 5 \]
Thus, the coordinates of vertex \( B \) are \( B(-5, 5) \).

Now, for the midpoint \( E \) of segment \( AC \):
\[ E = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) \]
Substituting the coordinates:
\[ (5, 2) = \left( \frac{1 + x_C}{2}, \frac{1 + y_C}{2} \right) \]
From the x-coordinates:
\[ 5 = \frac{1 + x_C}{2} \implies 10 = 1 + x_C \implies x_C = 9 \]
From the y-coordinates:
\[ 2 = \frac{1 + y_C}{2} \implies 4 = 1 + y_C \implies y_C = 3 \]
Thus, the coordinates of vertex \( C \) are \( C(9, 3) \).

### Step 2: Calculate the coordinates of the centroid \( G \) of triangle \( ABC \).

The formula for the centroid \( G \) of a triangle with vertices at \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) 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 vertices \( A(1, 1) \), \( B(-5, 5) \), and \( C(9, 3) \):
\[ G = \left( \frac{1 + (-5) + 9}{3}, \frac{1 + 5 + 3}{3} \right) \]
Calculating the x-coordinate:
\[ G_x = \frac{1 - 5 + 9}{3} = \frac{5}{3} \]
Calculating the y-coordinate:
\[ G_y = \frac{1 + 5 + 3}{3} = \frac{9}{3} = 3 \]
Thus, the coordinates of the centroid \( G \) are:
\[ G\left(\frac{5}{3}, 3\right) \]

### Final Answer:
The centroid of the triangle is \( G\left(\frac{5}{3}, 3\right) \).

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