In this section several practical theorems about circles are stated and motivations regarding the proofs of some of these theorems are offered.

Theorem 19-1

A diameter perpendicular to a chord bisects the chord and the two arcs that the chord subtends.

Figure 19.11 illustrates Theorem 19-1. Observe the congruence of the minor arcs that the diameter subtends.

F IGURE 19.11

E XAMPLE 19.1: Theorem 19-1

Chord AB is 9 inches long and drawn perpendicular to diameter CD What are the lengths of segment AE and segment EB ?

F IGURE 19.12

Solution: Since diameter CD is perpendicular to chord AB , Theorem 19-1 maintains that segment AE and segment EB are equal in measure. Therefore,