Question

lesson practice problems 8 problems 1 this diagram is a straightedge and compass construction of a line perpendicular to line ab passing through point c. explain why it is helpful to construct points d and a to be the same distance from c.

Explanation:

Step1: Recall Perpendicular Bisector

The perpendicular bisector of a segment has points equidistant from the segment's endpoints. Here, we aim to construct a perpendicular to \( AB \) through \( C \).

Step2: Analyze Point Distances

By making \( A \) and \( D \) equidistant from \( C \) (i.e., \( AC = DC \)), we set \( C \) as the midpoint of \( AD \). When we construct circles (or arcs) with centers at \( A \) and \( D \) (with equal radii, since \( AC = DC \) and radii can be set appropriately), their intersection points (like \( E \)) will lie on the perpendicular bisector of \( AD \). Since \( C \) is the midpoint of \( AD \) and \( AD \) is on \( AB \), the line through \( E \) and \( C \) will be perpendicular to \( AB \) (by the perpendicular bisector theorem: a line through the midpoint of a segment, perpendicular to the segment, is the perpendicular bisector, and points on the perpendicular bisector are equidistant from the segment's endpoints). So, equal distances for \( A \) and \( D \) from \( C \) help define the midpoint, enabling the perpendicular bisector construction (and thus the perpendicular line to \( AB \) through \( C \)).

Answer:

Making \( A \) and \( D \) equidistant from \( C \) (so \( C \) is the midpoint of \( AD \)) allows constructing the perpendicular bisector of \( AD \) (via circle intersections). Since \( AD \) lies on \( AB \), this bisector is perpendicular to \( AB \) and passes through \( C \), helping create the desired perpendicular line.