Question 1: Construct the perpendicular bisector of a segment with endpoints at (2, 3) and (6, 7). Label the point where it intersects the segment.

Answer: Point (4, 5) is the midpoint; the bisector is perpendicular at this point.

Question 2: Calculate the length of the segment with endpoints at (1, 2) and (5, 6).

Answer: 5

Question 3: Plot the graph of y = 2x. Draw a segment between (1, 2) and (4, 8). Construct its perpendicular bisector.

Answer: Perpendicular bisector passes through midpoint (2.5, 5).

Question 4: Identify the point on the perpendicular bisector of segment AB (endpoints at (3, 4) and (7, 8)) that is also equidistant from both endpoints.

Answer: Point (5, 6) (the midpoint)

Question 5: Explain why any point on the perpendicular bisector of a segment is equidistant from the segment's endpoints.

Answer: Because the perpendicular bisector is the set of points at equal distances from both endpoints, by definition. It is the locus of points equidistant from the two points.

Question 6: A triangle has vertices at (2, 3), (6, 3), and (4, 7). Find the perpendicular bisector of the side from (2, 3) to (6, 3).

Answer: Midpoint at (4, 3); bisector is the vertical line x=4.

Question 7: Construct the perpendicular bisector of a segment with endpoints at (-1, -1) and (3, 3). Then, identify the point on this bisector that is also equidistant from both endpoints.

Answer: Midpoint at (1, 1); point (1, 1) is equidistant from both endpoints.

Question 8: In a real-world context, if two fire stations are 4 km apart, describe how the perpendicular bisector of the segment connecting them can help locate the optimal site for a new fire station to ensure equal distance from both stations.

Answer: The perpendicular bisector represents all points equidistant from both stations. Placing the new station anywhere on this line ensures equal distance from both, optimizing response times.

Question 9: A student draws a segment and its perpendicular bisector, but the bisector does not intersect the segment at its midpoint. Identify and explain the mistake.

Answer: The student may have constructed the bisector incorrectly, perhaps by drawing a perpendicular line through one endpoint instead of the midpoint.

Question 10: Given a segment with endpoints at (0, 0) and (8, 0), find the equation of its perpendicular bisector.

Answer: Midpoint at (4, 0); bisector is the vertical line x=4.

Question 11: Challenge: Prove that the perpendicular bisectors of the sides of a triangle meet at a single point (the circumcenter).

Answer: By constructing the perpendicular bisectors of two sides and showing they intersect at a point equidistant from all three vertices, which is the circumcenter. This is a classic proof involving the properties of congruent triangles and circle theorems.