Key Concepts and Formulas

• The circumcenter of a triangle is the intersection point of the perpendicular bisectors of the sides of the triangle.
• The midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$.
• The slope of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is $\frac{y_2 - y_1}{x_2 - x_1}$.
• If two lines are perpendicular, the product of their slopes is -1.
• The point-slope form of a line is $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a point on the line.

Step 1: Understanding the Problem and Strategy

We are given the coordinates of two vertices, $P(-2, 4)$ and $Q(4, -2)$, and the equation of the perpendicular bisector of side $PR$, which is $2x - y + 2 = 0$. We want to find the circumcenter of $\triangle PQR$. The circumcenter is the intersection of the perpendicular bisectors. Thus, we will find the equation of the perpendicular bisector of side $PQ$ and then solve the system of equations formed by the two perpendicular bisectors to find their intersection point, which is the circumcenter.

Step 2: Finding the Perpendicular Bisector of PQ

To find the equation of the perpendicular bisector of $PQ$, we need its midpoint and slope.

2a. Calculate the Midpoint of PQ: The midpoint $M$ of $PQ$ is given by: $M = \left( \frac{-2 + 4}{2}, \frac{4 + (-2)}{2} \right) = \left( \frac{2}{2}, \frac{2}{2} \right) = (1, 1)$ The midpoint of $PQ$ is $(1, 1)$. This point lies on the perpendicular bisector.

2b. Calculate the Slope of PQ: The slope $m_{PQ}$ of $PQ$ is given by: $m_{PQ} = \frac{-2 - 4}{4 - (-2)} = \frac{-6}{6} = -1$ The slope of $PQ$ is $-1$.

2c. Calculate the Slope of the Perpendicular Bisector of PQ: Since the perpendicular bisector is perpendicular to $PQ$, its slope $m_{\perp PQ}$ satisfies: $m_{\perp PQ} \cdot m_{PQ} = -1$ $m_{\perp PQ} \cdot (-1) = -1$ $m_{\perp PQ} = 1$ The slope of the perpendicular bisector is $1$.

2d. Write the Equation of the Perpendicular Bisector of PQ: Using the point-slope form with midpoint $(1, 1)$ and slope $1$: $y - 1 = 1(x - 1)$ $y - 1 = x - 1$ $y = x$ This is the equation of the perpendicular bisector of $PQ$.

Step 3: Using the Given Perpendicular Bisector of Side PR

The equation of the perpendicular bisector of $PR$ is given as: $2x - y + 2 = 0$

Step 4: Finding the Circumcenter by Intersection

We have two equations for the perpendicular bisectors:

1. Perpendicular bisector of $PQ$: $y = x$
2. Perpendicular bisector of $PR$: $2x - y + 2 = 0$

Substituting $y = x$ into the second equation: $2x - x + 2 = 0$ $x + 2 = 0$ $x = -2$ Since $y = x$, we have $y = -2$. Therefore, the intersection point (circumcenter) is $(-2, -2)$.

Step 5: Conclusion and Verification

The circumcenter of $\triangle PQR$ is $(-2, -2)$, which corresponds to option (D).

Common Mistakes & Tips:

• Double-check the midpoint and slope formulas to avoid errors.
• Remember the relationship between the slopes of perpendicular lines.
• Carefully substitute and solve the system of equations to find the intersection point.

Summary:

We found the circumcenter of $\triangle PQR$ by finding the equation of the perpendicular bisector of side $PQ$, using the given equation of the perpendicular bisector of side $PR$, and then solving the system of equations to find their intersection point. The circumcenter is $(-2, -2)$.

The final answer is \boxed{(-2, -2)}, which corresponds to option (D).