Question

6 practice 6 (from unit 1, lesson 3) in this diagram, the segment cd is the perpendicular bisector of the segment ab. assume the conjecture that the set of points equidistant from a and b is the perpendicular bisector of ab is true. ab ⊥ cd
diagram with points a, m, b on one line, c, m, d on another, right angle at m, and e above
select all statements that must be true.
a am = bm
b cm = dm

Explanation:

To solve this, we analyze each option using the definition of a perpendicular bisector: a line (or segment) that is perpendicular to another segment and divides it into two equal parts.

Analyzing Option A: \( AM = BM \)

Since \( CD \) is the perpendicular bisector of \( AB \), by definition, \( M \) (the intersection point) divides \( AB \) into two equal segments. Thus, \( AM = BM \) must be true.

Analyzing Option B: \( CM = DM \)

The diagram shows \( M \) as the intersection of \( AB \) and \( CD \), but there is no information (e.g., \( M \) being the midpoint of \( CD \)) to confirm \( CM = DM \). The perpendicular bisector definition only guarantees \( AM = BM \) (for \( AB \)), not \( CM = DM \) (for \( CD \)) unless additional info is given (which it is not here).

• Option A: \( CD \) is the perpendicular bisector of \( AB \), so \( M \) splits \( AB \) into equal parts (\( AM = BM \)).
• Option B: No info confirms \( M \) is the midpoint of \( CD \), so \( CM = DM \) is not guaranteed.

Answer:

A. \( AM = BM \)