Question

select the correct answer from the drop - down menu.
the diagram shows a stage in the construction of a line perpendicular to \\( \overline { g h } \\) through point a.
the next step in this construction is

• to set the compass width to line segment ga and draw arcs centered at h across both sides of the line
• to use sufficient compass width and draw an arc centered at a crossing the previous arc
• to set the compass width to line gh and draw an arc centered at a crossing the previous arc in the image
• to keep the same compass width and draw an arc centered at h crossing the previous arc in the image

Explanation:

Step1: Analyze the construction of a perpendicular line

When constructing a perpendicular line to a given line through a point, after some initial arcs (as shown in the diagram), the standard procedure is to use the same compass width (the width used for the first - drawn arc) and draw an arc from the other point (in this case, from point \(H\)) that will intersect the previous arc. This is based on the SSS (Side - Side - Side) congruence criterion for triangles which is used in geometric constructions.

Step2: Evaluate each option

• Option 1: Setting the compass width to \(GA\) and drawing arcs centered at \(H\) is not the standard step. The compass width should be consistent with the initial arc - drawing width (not necessarily \(GA\)).
• Option 2: Using a "sufficient" (vague term) compass width and drawing an arc centered at \(A\) is incorrect. The next step should be from the other non - point of construction (\(H\) in this case) with the same width as the first arc.
• Option 3: Setting the compass width to \(GH\) is wrong. The compass width is related to the arc - drawing process (usually the distance between two points used in the initial arc construction, not the length of the line segment \(GH\)).
• Option 4: Keeping the same compass width (the width used for the first arc) and drawing an arc centered at \(H\) crossing the previous arc is in line with the geometric construction of a perpendicular line.

Answer:

to keep the same compass width and draw an arc centered at \(H\) crossing the previous arc in the image