Key Concepts and Formulas

• Midpoint Formula: The midpoint $M(x_m, y_m)$ of a line segment joining points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ has coordinates: $x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2}$
• Centroid Formula: The centroid $G(x_G, y_G)$ of a triangle with vertices $A(x_A, y_A)$, $B(x_B, y_B)$, and $C(x_C, y_C)$ has coordinates: $x_G = \frac{x_A + x_B + x_C}{3}, \quad y_G = \frac{y_A + y_B + y_C}{3}$

Step-by-Step Solution

Step 1: Define the triangle and given points.

• Why this step? To clearly define the problem and assign variables to the given information.
• Let the triangle be $ABC$ with vertex $A = (1, 1)$.
• Let $M_1 = (-1, 2)$ be the midpoint of side $AB$.
• Let $M_2 = (3, 2)$ be the midpoint of side $AC$.
• We want to find the centroid $G(x_G, y_G)$ of triangle $ABC$.

Step 2: Find the coordinates of vertex B.

• Why this step? To use the centroid formula, we need the coordinates of all three vertices. We know $A$, and we can find $B$ using the midpoint formula with $A$ and $M_1$.
• Let $B = (x_B, y_B)$. Since $M_1(-1, 2)$ is the midpoint of $AB$: $x_{M_1} = \frac{x_A + x_B}{2} \implies -1 = \frac{1 + x_B}{2}$ $-2 = 1 + x_B \implies x_B = -3$ $y_{M_1} = \frac{y_A + y_B}{2} \implies 2 = \frac{1 + y_B}{2}$ $4 = 1 + y_B \implies y_B = 3$
• So, the coordinates of vertex $B$ are $(-3, 3)$.

Step 3: Find the coordinates of vertex C.

• Why this step? We need $C$ to calculate the centroid. We use the midpoint formula with $A$ and $M_2$.
• Let $C = (x_C, y_C)$. Since $M_2(3, 2)$ is the midpoint of $AC$: $x_{M_2} = \frac{x_A + x_C}{2} \implies 3 = \frac{1 + x_C}{2}$ $6 = 1 + x_C \implies x_C = 5$ $y_{M_2} = \frac{y_A + y_C}{2} \implies 2 = \frac{1 + y_C}{2}$ $4 = 1 + y_C \implies y_C = 3$
• So, the coordinates of vertex $C$ are $(5, 3)$.

Step 4: Calculate the coordinates of the Centroid G.

• Why this step? Now that we have all three vertices $A(1, 1)$, $B(-3, 3)$, and $C(5, 3)$, we can directly apply the centroid formula.
• Using the centroid formula: $x_G = \frac{x_A + x_B + x_C}{3} = \frac{1 + (-3) + 5}{3} = \frac{3}{3} = 1$ $y_G = \frac{y_A + y_B + y_C}{3} = \frac{1 + 3 + 3}{3} = \frac{7}{3}$
• Therefore, the centroid $G$ of the triangle is $\left(1, \frac{7}{3}\right)$.

Alternative Method (Direct Centroid Formula using Given Points)

• Why this step? This provides a faster method to calculate the centroid in this specific scenario.
• The coordinates of the centroid $G$ can be directly calculated as: $G = \left( \frac{x_A + 2x_{M_1} + 2x_{M_2} - 3x_A}{3}, \frac{y_A + 2y_{M_1} + 2y_{M_2} - 3y_A}{3} \right)$ $G = \left( \frac{-x_A + 2x_{M_1} + 2x_{M_2} }{3}, \frac{-y_A + 2y_{M_1} + 2y_{M_2} }{3} \right)$
• Substituting the values with $A(1, 1)$, $M_1(-1, 2)$, and $M_2(3, 2)$: $x_G = \frac{-1 + 2(-1) + 2(3)}{3} = \frac{-1 - 2 + 6}{3} = \frac{3}{3} = 1$ $y_G = \frac{-1 + 2(2) + 2(2)}{3} = \frac{-1 + 4 + 4}{3} = \frac{7}{3}$
• This also yields the centroid $G = \left(1, \frac{7}{3}\right)$.

Common Mistakes & Tips:

• Carefully read the problem statement to correctly identify which vertex and midpoints are given. Misinterpreting the given information will lead to an incorrect setup.
• Double-check your arithmetic, especially when substituting negative values into the midpoint and centroid formulas.
• Remember the direct formula to quickly find the centroid when given one vertex and the midpoints of the sides originating from that vertex.

Summary

To find the centroid of the triangle, we first found the coordinates of all three vertices using the midpoint formula. Then, we applied the centroid formula to find the coordinates of the centroid. An alternative, more direct method was also presented. Both methods yield the same centroid coordinates, which is $\left(1, \frac{7}{3}\right)$.

Final Answer The final answer is $\boxed{\left( { 1,{7 \over 3}} \right)}$, which corresponds to option (C).