Question 1: Construct a perpendicular from point P to line l using a compass. Mark the foot of the perpendicular as point Q.

Answer: Construction steps: Draw arcs from P intersecting l, then from those intersections, draw arcs to locate Q.

Question 2: Given point P and line l, use a compass to construct a perpendicular from P to l. Measure the length of PQ.

Answer: Length of PQ (student to measure and record)

Question 3: Construct a perpendicular from point P to line l when P lies outside the segment, ensuring the foot Q lies on the extension of l.

Answer: Use compass to draw a perpendicular extension; mark Q on the extension line.

Question 4: Explain why the construction of a perpendicular from a point to a line is unique when the point is fixed and the line is given.

Answer: Because only one perpendicular can be drawn from a point to a line passing through the foot, ensuring the shortest distance.

Question 5: Construct a perpendicular from point P to line l, where P is inside the triangle formed by points A, B, and C. Use the compass to ensure accuracy.

Answer: Procedure: From P, draw arcs intersecting l; connect to form the perpendicular.

Question 6: A factory needs to position equipment at point P so that it is directly perpendicular to a main conveyor line l. Use compass construction to find the shortest distance from P to l.

Answer: Construct the perpendicular; measure PQ as the shortest distance.

Question 7: Construct a perpendicular from P to l and then extend the line beyond Q to form a right angle with the extension. Show all steps.

Answer: Construct perpendicular; extend line beyond Q; verify right angles.

Question 8: Analyze a construction where two students both attempt to construct a perpendicular from P to l but get different results. Identify the common mistake and correct it.

Answer: Common mistake: Drawing an arc from P that does not intersect l properly. Correction: Use correct compass radius and intersect l at two points.

Question 9: Using a compass, construct a perpendicular from point P to line l that is at a 45° angle to a given segment AB on the same plane. Explain your process.

Answer: Construct perpendicular from P; measure a 45° angle using the compass and protractor; verify the angle.

Question 10: Challenge: Construct a triangle where the perpendiculars from two vertices to the opposite sides are equal in length. Use compass to verify if the construction is correct.

Answer: Construct the triangle; draw perpendiculars; compare PQ and PR lengths for equality.

Question 11: Construct a perpendicular from P to l, then reflect P across the line of the perpendicular to find P'. Confirm P' lies on the same perpendicular line and measure the distance between P and P'.

Answer: Construction steps: Reflect P; verify collinearity; measure PP'.

Question 12: Identify a common error in compass construction of a perpendicular and explain how to avoid it.

Answer: Forgetting to set the compass radius correctly; ensure the same radius is used for all arcs.