Question

constructions

1. carla drew two arcs of the same radius by placing her compass at each endpoint of \\( \overline { a b } \\) to produce

the picture below.
if she connects the two points where the arcs intersect with a
straightedge, what must be true about the segment she creates?
a. it is congruent to and perpendicular to \\( \overline { a b } \\).
b. it is congruent to and bisects \\( \overline { a b } \\).
c. it is perpendicular to and bisects \\( \overline { a b } \\).
d. it is congruent to, perpendicular to, and bisects \\( \overline { a b } \\).

Explanation:

Step1: Recall the property of perpendicular bisector construction

When we draw two arcs of the same radius with centers at the endpoints of a line segment (\(\overline{AB}\) in this case), the line joining the intersection points of the arcs is the perpendicular bisector of the line segment \(\overline{AB}\).

Step2: Analyze the properties of the perpendicular bisector

A perpendicular bisector of a line segment \(\overline{AB}\) has two properties:

1. It is perpendicular to \(\overline{AB}\). Mathematically, if the slope of \(\overline{AB}\) is \(m_1\), and the slope of the bisector is \(m_2\), then \(m_1\times m_2=- 1\).
2. It bisects \(\overline{AB}\), which means it divides \(\overline{AB}\) into two equal - length sub - segments. Let the mid - point of \(\overline{AB}\) be \(M\). If \(A(x_1,y_1)\) and \(B(x_2,y_2)\), then \(M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\) and the length from \(A\) to \(M\) is equal to the length from \(M\) to \(B\).

Answer:

C. It is perpendicular to and bisects \(\overline{AB}\).