Question

1. the diagram to the right was constructed with a compass, straight edge, and pencil. a, c, and e are each the center of a circle, and c is the center of the other circle. select all the true statements. a) $overline{ab}congoverline{ad}$ b) $overline{ac}congoverline{ad}$ c) $overline{eb}congoverline{ed}$ d) $overline{ac}congoverline{ce}$ e) $overline{eb}congoverline{ec}$

Explanation:

Step1: Recall circle - radius property

In a circle, all radii are congruent.

Step2: Analyze option A

Since points \(A\), \(C\), and \(E\) are circle - centers. For circle with center \(A\), \(AB\) and \(AD\) are radii of circle \(A\). So, \(\overline{AB}\cong\overline{AD}\).

Step3: Analyze option B

\(\overline{AC}\) is not a radius of the circle with center \(A\) and \(\overline{AD}\) is a radius of circle \(A\). They are not necessarily congruent.

Step4: Analyze option C

For circle with center \(E\), \(EB\) and \(ED\) are radii of circle \(E\). So, \(\overline{EB}\cong\overline{ED}\).

Step5: Analyze option D

\(\overline{AC}\) and \(\overline{CE}\) are radii of the circles centered at \(A\) and \(E\) respectively. Since the circles are constructed in a way that they are related geometrically and \(AC\) and \(CE\) are part of the equal - distance construction, \(\overline{AC}\cong\overline{CE}\).

Step6: Analyze option E

\(\overline{EB}\) is a radius of circle \(E\) and \(\overline{EC}\) is a radius of circle \(E\). So, \(\overline{EB}\cong\overline{EC}\).

Answer:

A. \(\overline{AB}\cong\overline{AD}\)
C. \(\overline{EB}\cong\overline{ED}\)
D. \(\overline{AC}\cong\overline{CE}\)
E. \(\overline{EB}\cong\overline{EC}\)