Question

joaquin is constructing the perpendicular bisector of \\( \overline { a b } \\). he opens his compass so that the distance from the two points of the compass is wider than half the length of \\( \overline { a b } \\). he then places the point of the compass on point a and draws an arc across \\( \overline { a b } \\). what is his next step?

a. place the point of the compass on point b and mark where the arc intersects \\( \overline { a b } \\).
b. place the point of the compass on the midpoint of \\( \overline { a b } \\) and mark the points directly above and below the midpoint.
c. place the point of the compass on point b and draw an arc across \\( \overline { a b } \\).
d. place the point of the compass on the midpoint of \\( \overline { a b } \\) and draw an arc in either direction.

Explanation:

Step1: Recall perpendicular bisector construction steps

To construct the perpendicular bisector of a line segment \(AB\), after drawing an arc from point \(A\) with the compass width greater than half of \(AB\), the next step is to use the same compass width and draw an arc from point \(B\). This is because we need two intersection points of the arcs (one from \(A\) and one from \(B\)) to draw the perpendicular bisector.

Step2: Analyze each option

• Option A: Marking where the arc intersects \(AB\) from \(B\) (with the current description) is not the standard next - step. We need to draw an arc (not just mark an intersection on \(AB\)) to get intersection points above and below \(AB\).
• Option B: We don't know the mid - point yet. The process of constructing the perpendicular bisector is to find the mid - point (by intersecting arcs) first.
• Option C: This is the correct next step. Using the same compass width (set when we were at point \(A\)), place the compass at point \(B\) and draw an arc across \(AB\). The intersection points of the two arcs (one from \(A\) and one from \(B\)) will help us draw the perpendicular bisector.
• Option D: Again, we don't know the mid - point at this stage of construction.

Answer:

C. Place the point of the compass on point \(B\) and draw an arc across \(\overline{AB}\).