Question

you are given the steps for using a compass and a straightedge to construct a line perpendicular to a given line through a point not lying on the given line. arrange the steps in the proper sequence.
draw an arc cutting the given line at two points. mark and label the points of intersection p and q.
move the compass needle to p and make an arc below the line. keep the compass width, and move the compass needle to q. draw an arc below the line crossing the other arc. mark and label the point of intersection s.
place the compass needle on the external point r. make the compass width greater than the distance from r to the given line.
draw a line from the external point r to the point where the arcs intersect, s. line rs is perpendicular to line pq and passes through the external point r.

Explanation:

Step1: Place the compass needle on the external point \( R \). Make the compass width greater than the distance from \( R \) to the given line.

This step sets up the compass for the first arc - drawing.

Step2: Draw an arc cutting the given line at two points. Mark and label the points of intersection \( P \) and \( Q \).

This creates two intersection points on the given line.

Step3: Move the compass needle to \( P \) and make an arc below the line. Keep the compass width, and move the compass needle to \( Q \). Draw an arc below the line crossing the other arc. Mark and label the point of intersection \( S \).

This step constructs the second set of arcs to find the intersection point \( S \).

Step4: Draw a line from the external point \( R \) to the point where the arcs intersect, \( S \). Line \( RS \) is perpendicular to line \( PQ \) and passes through the external point \( R \).

This final step draws the perpendicular line.

Answer:

1. Place the compass needle on the external point \( R \). Make the compass width greater than the distance from \( R \) to the given line.
2. Draw an arc cutting the given line at two points. Mark and label the points of intersection \( P \) and \( Q \).
3. Move the compass needle to \( P \) and make an arc below the line. Keep the compass width, and move the compass needle to \( Q \). Draw an arc below the line crossing the other arc. Mark and label the point of intersection \( S \).
4. Draw a line from the external point \( R \) to the point where the arcs intersect, \( S \). Line \( RS \) is perpendicular to line \( PQ \) and passes through the external point \( R \).