Question

which of the following is the minor arc of the circle shown below?
a. $overarc{atb}$

b. $overarc{ab}$

c. $overarc{abt}$

d. $overarc{to}$

Explanation:

Step1: Recall minor arc definition

A minor arc is an arc of a circle whose measure is less than \(180^\circ\), and it is the shorter arc between two points on a circle.

Step2: Analyze each option

• Option A: \(\overarc{ATB}\) – This arc goes through \(T\), likely the major arc (longer path) as the central angle for the other arc (\(\overarc{AB}\)) is \(100^\circ\), so this would be the longer arc (measure \(360 - 100=260^\circ\), more than \(180^\circ\)).
• Option B: \(\overarc{AB}\) – The central angle \(\angle AOB = 100^\circ\), so the measure of arc \(AB\) is \(100^\circ\), which is less than \(180^\circ\), so it is a minor arc.
• Option C: \(\overarc{ABT}\) – Similar to option A, this is a major arc (longer path) as it includes more than a semicircle.
• Option D: \(\overarc{TO}\) – \(O\) is the center, not a point on the circumference (arcs are between points on the circle), so this is not a valid arc of the circle.

Answer:

B. \(\overarc{AB}\)