Question 1: Using a compass, construct a perpendicular line from the point P to the given line l. Describe your steps.

Answer: Construct the perpendicular by drawing arcs from P intersecting l, then drawing arcs from those points to find the perpendicular intersection.

Question 2: Measure the length of the perpendicular segment from point P to line l using your compass and ruler.

Answer: Students will perform the measurement based on their construction.

Question 3: Construct a perpendicular from point P to line l that is exactly 5 cm long. Show the construction process.

Answer: Adjust the compass to 5 cm, then perform the perpendicular construction ensuring the segment is exactly 5 cm.

Question 4: Plot the graph of y=2x and construct a perpendicular from the point (3,6) to this line. Show all steps.

Answer: Students plot y=2x and then use compass and straightedge to draw the perpendicular from (3,6).

Question 5: Construct an equilateral triangle with side length 6 cm using only a compass. Then, from the vertex opposite the side, construct a perpendicular to that side.

Answer: Students perform the construction and then draw the perpendicular from the vertex to the side.

Question 6: In a given figure, a point P outside line l is 7 cm from l. Construct the perpendicular from P to l and verify it is perpendicular.

Answer: Construction steps, including arcs and verification of right angles.

Question 7: An error is made in constructing a perpendicular: the arcs from P do not intersect l correctly. Identify and explain the mistake.

Answer: Common mistake: arcs from P not accurately intersecting, or incorrect compass setting.

Question 8: Construct the perpendicular from point P to line l such that the segment is twice as long as the original perpendicular. Show the construction.

Answer: Adjust the compass to twice the length of the initial perpendicular and re-construct accordingly.

Question 9: Using a compass, construct two different perpendiculars from point P to line l and compare their lengths. Explain the reason for any differences or similarities.

Answer: Perpendiculars from the same point to the same line are equal; any differences indicate a construction error.

Question 10: Challenge: Given a line l and a point P not on l, construct the shortest possible perpendicular from P to l. Explain why your construction is minimal.

Answer: Construct the perpendicular as usual; the shortest segment is the one perpendicular to l through P, which is guaranteed by the construction.