Question

select the correct answer.
what is the first step in the construction of a line perpendicular to \\( \overleftrightarrow { a b } \\) and passing through an external point \\( m \\) ? assume that the construction uses only a compass and a straightedge.
\\( \bigcirc \\) a. place the compass needle on \\( m \\) and, keeping the compass width the same, draw two arcs that intersect \\( \overleftrightarrow { a b } \\)
\\( \bigcirc \\) b. place the compass needle at any point on \\( \overleftrightarrow { a b } \\), and set the compass width to \\( m \\).
\\( \bigcirc \\) c. draw a line from point \\( m \\) passing through \\( \overleftrightarrow { a b } \\), and mark the point of intersection.
\\( \bigcirc \\) d. place the compass needle at any two points on \\( \overleftrightarrow { a b } \\), and draw two arcs through \\( m \\).

Explanation:

Step1: Recall the construction steps

When constructing a perpendicular line from an external point \(M\) to a line \(\overleftrightarrow{AB}\) using a compass and straight - edge, the first step is to use the compass centered at the external point \(M\).

Step2: Analyze each option

• Option A: Place the compass needle on \(M\) and, keeping the compass width the same, draw two arcs that intersect \(\overleftrightarrow{AB}\). This is the correct first step as it starts the process of creating congruent segments to help in constructing the perpendicular.
• Option B: Placing the compass needle at any point on \(\overleftrightarrow{AB}\) and setting the width to \(M\) is not the first step. The construction should start from the external point \(M\) for the perpendicular line construction.
• Option C: Drawing a line from \(M\) through \(\overleftrightarrow{AB}\) is not the first step. We need to use the compass first to create reference points on \(\overleftrightarrow{AB}\).
• Option D: Placing the compass needle at any two points on \(\overleftrightarrow{AB}\) and drawing arcs through \(M\) is not the first step. The first action should be centered at the external point \(M\).

Answer:

A. Place the compass needle on \(M\) and, keeping the compass width the same, draw two arcs that intersect \(\overleftrightarrow{AB}\)