Question

1. line segment ab is shown in the diagram below. which two sets of construction marks, labeled i, ii, iii, and iv, are part of the construction of the perpendicular bisector of line segment ab? 1) i and ii 2) i and iii 3) iii and iii 4) ii and iv 2) the diagram below shows the construction of the perpendicular bisector of ab. which statement is not true? 1) ac = cb 2) cb = 1/2 ab 3) ac = 2ab 4) ac + cb = ab 3) based on the construction below, which conclusion is not always true? 1) ab ⊥ cd 2) ab = cd 3) ae = eb 4) ce = de

Explanation:

Step1: Recall perpendicular - bisector construction

In the construction of the perpendicular bisector of a line - segment \(AB\), we use a compass to draw arcs of equal radius from points \(A\) and \(B\). The intersection points of these arcs are used to draw the perpendicular bisector. The arcs from \(A\) and \(B\) create the necessary construction marks.

Step2: Analyze question 1

For the construction of the perpendicular bisector of line - segment \(AB\), the arcs drawn from \(A\) and \(B\) (corresponding to the construction marks) are crucial. The sets of marks that are part of the construction are the ones from the compass - arcs centered at \(A\) and \(B\). Looking at the first question, the correct sets of construction marks for the perpendicular bisector of \(AB\) are the ones from the arcs centered at \(A\) and \(B\), which are I and III.

Step3: Analyze question 2

In the construction of the perpendicular bisector of \(\overline{AB}\), the point \(C\) is the mid - point of \(\overline{AB}\). So, \(AC = CB=\frac{1}{2}AB\) and \(AC + CB=AB\). The statement \(AC = 2AB\) is not true.

Step4: Analyze question 3

In the construction of the perpendicular bisector of a line segment, the perpendicular bisector \(\overline{CD}\) of \(\overline{AB}\) implies \(\overline{AB}\perp\overline{CD}\), \(AE = EB\) (since \(E\) is the mid - point of \(AB\) on the perpendicular bisector), and \(CE = DE\) (due to the symmetry of the construction). However, there is no reason for \(AB\) and \(CD\) to be equal in length. So, the conclusion \(AB = CD\) is not always true.

Answer:

1. 2) I and III
2. 3) \(AC = 2AB\)
3. 2) \(AB = CD\)