Question

marcello is constructing the perpendicular bisector of \\( \overline { a b } \\). he has already constructed arcs as shown. what should marcello do for his next step? place the point of the compass on point \\( x \\) and draw an arc, using \\( a y \\) as the width for the opening of the compass. use the straightedge to draw a line through points \\( x \\) and \\( y \\). place the point of the compass on point \\( x \\) and draw an arc, using \\( a x \\) as the width for the opening of the compass. use the straightedge to draw \\( \overleftrightarrow { a x } \\) and \\( \overleftrightarrow { a y } \\)

Explanation:

When constructing the perpendicular bisector of a line segment, after drawing the arcs from both endpoints (which gives points \(X\) and \(Y\) in this case), the next step is to connect these two intersection points of the arcs. A line drawn through \(X\) and \(Y\) will be the perpendicular bisector of \(\overline{AB}\).

• The first option about using \(AY\) as the compass width is incorrect because we don't need to draw another arc in that way for the perpendicular bisector construction.
• The third option about using \(AX\) as the compass width is also not the next step in constructing the perpendicular bisector (we already have the intersection points \(X\) and \(Y\) from previous arc - drawing steps).
• The fourth option of drawing \(\overleftrightarrow{AX}\) and \(\overleftrightarrow{AY}\) is not relevant to constructing the perpendicular bisector of \(\overline{AB}\).

Answer:

Use the straightedge to draw a line through points \(X\) and \(Y\)