Question

a) the construction of creating the perpendicular bisector of \\( \overline { p q } \\) is started below. how would the construction be different if you changed the compass setting in the next step of the perpendicular bisector construction?

Explanation:

When constructing a perpendicular bisector of a line segment \(\overline{PQ}\), the standard compass - setting is to have the compass width greater than half of the length of \(\overline{PQ}\). If you change the compass setting:

• If the compass setting is less than half of the length of \(\overline{PQ}\), the arcs drawn from points \(P\) and \(Q\) will not intersect each other. As a result, you will not be able to find the two intersection points that are used to draw the perpendicular bisector.
• If the compass setting is the same as the previous step (already greater than half of the length of \(\overline{PQ}\)), the construction will still work, but the intersection points of the arcs (used to draw the perpendicular bisector) will be in different positions. The perpendicular bisector will still be valid, but the visual appearance of the arcs (their size) will change.

Answer:

If the compass setting is less than half of \(PQ\), the arcs from \(P\) and \(Q\) will not intersect, and the perpendicular bisector cannot be constructed. If the compass setting is still greater than half of \(PQ\) (but different from the previous non - half length setting), the intersection points of the arcs (used for constructing the perpendicular bisector) will change position, but the perpendicular bisector will still be valid.