Question

1. this diagram is a straightedge and compass construction. a is the center of one circle, and b is the center of the other.

a. name a pair of perpendicular line segments.
b. name a pair of line segments with the same length.
(from unit 1, lesson 3.)

1. a, b, and c are the centers of the 3 circles. select all the segments that are congruent to ab.

a. hf
b. ha
c. ce
d. cd
e. bd
f. bf
(from unit 1, lesson 4.)

Explanation:

Step1: Analyze perpendicular segments in first - diagram

In a straight - edge and compass construction of two intersecting circles with centers \(A\) and \(B\), the line joining the intersection points of the two circles is perpendicular to the line segment \(AB\). So, \(AB\) and the line segment joining the two intersection points (let's call it \(CD\)) are perpendicular.

Step2: Analyze equal - length segments in first - diagram

The radii of the same circle are equal. If we consider the circle with center \(A\), \(AM\) and \(AN\) (where \(M\) and \(N\) are points on the circle centered at \(A\)) are equal.

Step3: Analyze congruent segments in second - diagram

Since \(A\), \(B\), and \(C\) are the centers of the circles and the construction is symmetric, \(AB = BC=CA\). Also, \(HA = HB=HC = FA=FB = FC\). Segments \(HA\) and \(AB\) are congruent.

Answer:

a. \(AB\) and the line segment joining the two intersection points of the circles in the first diagram
b. \(AM\) and \(AN\) (radii of the circle centered at \(A\) in the first diagram)

1. B. \(HA\)