Question

lesson 5 practice problems

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

select all true statements.
a. line ef is the bisector of angle bac.
b. line ef is the perpendicular bisector of segment ba.
c. line ef is the perpendicular bisector of segment ac.
d. line ef is the perpendicular bisector of segment bd.
e. line ef is parallel to line cd

Explanation:

Step1: Recall perpendicular - bisector property

In a perpendicular - bisector construction, if two points (like \(A\) and \(D\)) are equidistant from a third point (\(C\)), then the line passing through the intersection of the circles centered at \(A\) and \(D\) (the perpendicular line) is the perpendicular - bisector of the segment \(AD\). When constructing a perpendicular line to \(AB\) through \(C\), making \(CA = CD\) helps in creating an isosceles triangle \(\triangle ACD\). The perpendicular bisector of the base \(AD\) of an isosceles triangle \(\triangle ACD\) passes through the vertex \(C\) and is perpendicular to \(AB\).

Step2: Analyze second problem

For the second problem, in a straight - edge and compass construction of an angle bisector, the line \(EF\) is constructed such that it divides the angle \(\angle BAC\) into two equal angles. By the properties of straight - edge and compass construction of an angle bisector, the intersection of the arcs (centered at the points where the initial arcs from \(A\) intersect the sides of the angle) gives the angle - bisector. Also, since the construction is symmetric with respect to the angle bisector, the line \(EF\) is the perpendicular bisector of the segment joining the points of intersection of the arcs on the sides of the angle. Here, the points of intersection of the arcs on the sides of \(\angle BAC\) are symmetric about \(EF\), and \(EF\) is the perpendicular bisector of the segment joining these two points. In this case, the segment is \(BD\).

Answer:

1. Making \(CA = CD\) helps in creating an isosceles triangle \(\triangle ACD\). The perpendicular bisector of the base \(AD\) of an isosceles triangle \(\triangle ACD\) passes through the vertex \(C\) and is perpendicular to \(AB\).
2. A. Line \(EF\) is the bisector of angle \(BAC\); D. Line \(EF\) is the perpendicular bisector of segment \(BD\)