Question 1: Given points A(2, 3) and B(8, 7), find the midpoint of AB. Then, construct the perpendicular bisector of AB on the grid.

Answer: Midpoint (5, 5); Perpendicular bisector passes through (5, 5) and is perpendicular to AB.

Question 2: Prove that any point P on the perpendicular bisector of segment CD is equidistant from C and D, using coordinate geometry.

Answer: Let C(x1, y1), D(x2, y2), P(x, y). Show that distance PC equals distance PD by calculating (x - x1)^2 + (y - y1)^2 and (x - x2)^2 + (y - y2)^2, and demonstrating equality when P lies on the perpendicular bisector.

Question 3: Construct a triangle with vertices at A(1, 2), B(7, 2), and C(4, 6). Find and draw the perpendicular bisectors of AB and AC. Identify their intersection point.

Answer: Perpendicular bisector of AB: passes through midpoint (4,2) and is vertical; of AC: passes through midpoint (2.5, 4), slope calculation, intersection point.

Question 4: In a real-world scenario, a park ranger wants to set up a fire station exactly halfway between two lakes located at (3, 4) and (9, 10). Describe how to locate this optimal site using the perpendicular bisector.

Answer: Calculate the midpoint of the lakes, then draw the perpendicular bisector of the segment connecting them. The fire station should be located anywhere on this bisector.

Question 5: Given a segment EF with endpoints E(2, 5) and F(6, 9), determine the equation of the perpendicular bisector of EF.

Answer: Midpoint at (4, 7), slope of EF is 1, perpendicular slope is -1, equation: y - 7 = -1(x - 4).

Question 6: Explain why the perpendicular bisector of a segment is also a locus of points equidistant from the endpoints of that segment.

Answer: Any point on the perpendicular bisector maintains equal distances to both endpoints by the definition of the locus, as demonstrated through geometric or algebraic proof.

Question 7: Construct and label the perpendicular bisectors of two sides of a given triangle ABC. Verify that their intersection point is the triangle's circumcenter.

Answer: Draw bisectors, find their intersection, confirm equidistance from the vertices to the intersection point.

Question 8: A student mistakenly draws the perpendicular bisector of a segment but fails to bisect the segment accurately. Identify and explain the possible error in their construction.

Answer: Error: The student may have drawn a line perpendicular to the segment but not through its midpoint. The bisector must pass through the segment's midpoint.

Question 9: Challenge: Given points G(1, 3) and H(9, 7), find the equation of the perpendicular bisector of GH. Then, prove that any point on this line is equidistant from G and H.

Answer: Midpoint (5, 5), slope of GH is 0.5, perpendicular slope -2, equation: y - 5 = -2(x - 5). Show that distance formulas are equal for any point on this line.

Question 10: Explain how the concept of the perpendicular bisector can be used to find the center of a circle passing through three non-collinear points.

Answer: Construct perpendicular bisectors of any two sides of the triangle formed by the three points. Their intersection is the circle's center, equidistant from all three points.