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.

a. place the compass needle on $ m $ and, keeping the compass width the same, draw two arcs that intersect $ \overleftrightarrow { a b } $.
b. place the compass needle at any point on $ \overleftrightarrow { a b } $, and set the compass width to $ m $.
c. draw a line from point $ m $ passing through $ \overleftrightarrow { a b } $, and mark the point of intersection.
d. place the compass needle at any two points on $ \overleftrightarrow { a b } $, and draw two arcs through $ m $.

Question

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.

a. place the compass needle on $ m $ and, keeping the compass width the same, draw two arcs that intersect $ \overleftrightarrow { a b } $.
b. place the compass needle at any point on $ \overleftrightarrow { a b } $, and set the compass width to $ m $.
c. draw a line from point $ m $ passing through $ \overleftrightarrow { a b } $, and mark the point of intersection.
d. place the compass needle at any two points on $ \overleftrightarrow { a b } $, and draw two arcs through $ m $.

Accepted Answer

# Brief Explanations:
When constructing a perpendicular line from an external point $M$ to $\overleftrightarrow{AB}$ using a compass and straight - edge, the first step is to create two intersection points on $\overleftrightarrow{AB}$ from the external point $M$. Placing the compass needle on $M$ and drawing two arcs (with the same compass width) that intersect $\overleftrightarrow{AB}$ gives us these two points. This is a fundamental step in the construction process as it allows us to use the properties of circles (from the compass arcs) to later construct the perpendicular.
- Option B: Setting the compass width to $M$ (a point) is not a valid operation. The compass width is set to a length, not a point.
- Option C: Drawing a random line from $M$ through $\overleftrightarrow{AB}$ does not start the construction of a perpendicular in the systematic (compass - and - straightedge) way.
- Option D: Placing the compass on two points on $\overleftrightarrow{AB}$ to draw arcs through $M$ is not the first step. We need to first establish the intersection of arcs (from $M$) with $\overleftrightarrow{AB}$ to define key points for the perpendicular construction.

# Answer:
A. Place the compass needle on $M$ and, keeping the compass width the same, draw two arcs that intersect $\overleftrightarrow{AB}$

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