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Step-by-step Solution
Step 1: Identify the Given Points
We have triangle PQR with coordinates:
• P at $(-2, 4)$
• Q at $(4, -2)$
Step 2: Find the Midpoint of PQ
The midpoint $M$ of the line segment PQ is given by
$$
M\Bigl(\dfrac{x_P + x_Q}{2},\, \dfrac{y_P + y_Q}{2}\Bigr).
$$
Substituting the coordinates:
$$
M\Bigl(\dfrac{-2 + 4}{2}, \dfrac{4 + (-2)}{2}\Bigr)
= M\bigl(1, 1\bigr).
$$
Step 3: Calculate the Slope of PQ
The slope of PQ is
$$
m_{PQ} = \dfrac{y_Q - y_P}{x_Q - x_P}
= \dfrac{-2 - 4}{4 - (-2)}
= \dfrac{-6}{6}
= -1.
$$
Step 4: Write the Equation of the Perpendicular Bisector of PQ
The slope of the perpendicular bisector of PQ will be the negative reciprocal of $m_{PQ}$.
Since $m_{PQ} = -1$, the perpendicular slope is $1$.
Using the point-slope form with the midpoint $M(1, 1)$:
$$
y - 1 = 1(x - 1).
$$
Simplifying:
$$
y - 1 = x - 1
\quad\Longrightarrow\quad
y = x.
$$
Hence, the perpendicular bisector of PQ is $y = x$.
Step 5: Use the Given Perpendicular Bisector of PR
The problem states that the perpendicular bisector of PR is given by:
$$
2x - y + 2 = 0.
$$
We need to find the point of intersection of
$$
y = x
\quad\text{and}\quad
2x - y + 2 = 0
$$
to get the circumcenter.
Step 6: Solve the Two Equations Simultaneously
Substitute $y = x$ into $2x - y + 2 = 0$:
$$
2x - x + 2 = 0
\quad\Longrightarrow\quad
x + 2 = 0
\quad\Longrightarrow\quad
x = -2.
$$
Since $y = x$, we also have $y = -2$.
Therefore, the intersection point (and hence the circumcenter) is $(-2, \,-2)$.
Final Answer
The centre of the circumcircle of the triangle PQR is
$$
\bigl(-2,\,-2\bigr).
$$
