Question

lesson 5 construction techniques 3. perpendicular lines and angle bisecton lets use tools to solve some construction challenges. warm-up two circles points a and b are each at the centers of circles of radius ab. 1 compare the distance ea to the distance eb. be prepared to explain your reasoning. 2 compare the distance fa to the distance fb. be prepared to explain your reasoning. 3 draw line ef, and write a conjecture about its relationship with segment ab.

Explanation:

Step1: Analyze the radii of the circles

Since \(A\) and \(B\) are centers of circles with radius \(AB\), for point \(E\) on both circles, \(EA = AB\) (radius of circle \(A\)) and \(EB=AB\) (radius of circle \(B\)). So \(EA = EB\).

Step2: Analyze the radii for point \(F\)

Similarly, for point \(F\) on both circles, \(FA = AB\) (radius of circle \(A\)) and \(FB = AB\) (radius of circle \(B\)). So \(FA=FB\).

Step3: Consider the properties of perpendicular bisectors

Points \(E\) and \(F\) are equidistant from \(A\) and \(B\). By the perpendicular - bisector theorem (a set of points equidistant from two endpoints of a segment lies on the perpendicular bisector of the segment), line \(EF\) is the perpendicular bisector of segment \(AB\).

Answer:

1. \(EA = EB\)
2. \(FA = FB\)
3. Line \(EF\) is the perpendicular bisector of segment \(AB\)