Question

1. in this diagram, line segment cd is the perpendicular bisector of line segment ab. assume the conjecture that the set of points equidistant from a and b is the perpendicular bisector of ab is true. select all statements that must be true.

a. am = bm
b. cm = dm
c. ea = em
d. ea < eb
e. am < ab
f. am > bm
(from unit 1, lesson 3.)

1. the diagram was constructed with straightedge and compass tools. name all segments that have the same length as segment ac.

(from unit 1, lesson 1.)

Explanation:

Step1: Recall property of perpendicular bisector

A perpendicular bisector of a line - segment divides the line - segment into two equal parts. Since \(CD\) is the perpendicular bisector of \(AB\), \(M\) is the mid - point of \(AB\). So \(AM = BM\). Also, \(AM=\frac{1}{2}AB\), so \(AM\lt AB\).

Step2: Analyze other statements

There is no information given to suggest that \(CM = DM\) (the perpendicular bisector of \(AB\) does not imply equal lengths for parts of \(CD\)), no reason for \(EA = EM\), and since \(E\) lies on the perpendicular bisector of \(AB\), \(EA=EB\).

For question 7, when constructing with a straightedge and compass, circles are drawn with the same radius. If we assume the circles are drawn with the same radius, the segments with the same length as \(AC\) are \(AD\), \(BC\), \(BD\), \(AE\), \(BE\) because they are all radii of congruent circles in the construction.

Answer:

1. A. \(AM = BM\), E. \(AM\lt AB\)
2. Segments with the same length as \(AC\) are \(AD\), \(BC\), \(BD\), \(AE\), \(BE\)