Answer:
(-2, 2)
Explanation:
You want the circumcenter of the triangle with vertices A(2, -3), B(2, 7), and C(-6, -3).
Circumcenter
The circumcenter is the center of the circle that circumscribes the triangle. Here, the triangle is a right triangle, so the circumcenter is the midpoint of the hypotenuse.
Midpoint
The midpoint of segment BC is the average of the coordinates of B and C:
X = (B +C)/2 = ((2, 7) +(-6, -3))/2 = (2 -6, 7 -3)/2 = (-4, 4)/2
X = (-2, 2)
The circumcenter is (-2, 2).
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Additional comment
The right angle subtends an arc of twice its measure, 180°, so the circumscribed triangle occupies a semicircle, and the hypotenuse is a diameter. That puts the center of the circle at the midpoint of the hypotenuse.