Question 1: Using a compass and straightedge, construct the perpendicular bisector of a given line segment AB, where point A is at (2, 3) and point B is at (8, 7).

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

Question 2: Construct the perpendicular bisector of a segment with endpoints at (1, 2) and (5, 6).

Answer: Midpoint at (3, 4); perpendicular bisector through (3, 4) with right angles.

Question 3: Given a line segment with endpoints at (0, 0) and (6, 0), draw its perpendicular bisector and state its equation.

Answer: Midpoint at (3, 0); perpendicular bisector is y = 0; but since it’s horizontal, the perpendicular bisector is x=3.

Question 4: Construct the perpendicular bisector of a segment that is not aligned horizontally or vertically. Describe the steps briefly.

Answer: Step 1: Draw the segment. Step 2: With the compass, set width more than half the segment length, arc above and below. Step 3: Repeat from both endpoints. Step 4: Connect intersection points of arcs to form the bisector.

Question 5: Identify and correct the mistake: A student constructs a perpendicular bisector but it does not pass through the midpoint of the segment. What went wrong?

Answer: They used different compass widths for arcs from each endpoint, so the intersection points are not aligned correctly. To fix, ensure same compass width and accurate arc intersections.

Question 6: Construct the perpendicular bisector of a segment with endpoints at (4, 2) and (10, 8). Then, find the coordinates of the point where it intersects the line y = 3x - 2.

Answer: Midpoint at (7, 5); perpendicular bisector through (7, 5); its equation x=7. Intersection with y=3x-2: y=3(7)-2=19, point (7,19).

Question 7: A real-world scenario: A surveyor needs to find the midpoint of a straight boundary line between two points at coordinates (3, 4) and (9, 10). Explain how compass construction can assist.

Answer: Use compass to find the midpoint by constructing the perpendicular bisector of the segment; the crossing point with other reference lines gives the midpoint.

Question 8: Given a segment of length 10 units, construct its perpendicular bisector and verify its correctness by measuring the distances from the midpoint to each endpoint.

Answer: Midpoint divides segment into two equal parts, each 5 units long; the perpendicular bisector passes through the midpoint, and distances from midpoint to endpoints are equal.

Question 9: Challenge: Construct a triangle where a given side is bisected perpendicularly using compass techniques, then extend this to bisect an angle at one vertex.

Answer: Construct the perpendicular bisector of the side for midpoint; to bisect the angle, use arc intersections from the vertex and the bisector to find the angle bisector.

Question 10: Identify the common mistake in constructing perpendicular bisectors when students try to use only a ruler without compass, and suggest the correction.

Answer: Ruler alone cannot accurately find equidistant points; the correction is to use compass arcs to locate equal distances reliably.