Question

what is the measure of (overarc{ac}) (the minor arc)?
diagram: circle with points a, b, c; angle at b is (62^circ)
options:
a. (62^circ)
b. (31^circ)
c. (236^circ)
d. (124^circ)

Explanation:

Step1: Identify triangle type

Triangle \( \triangle ABA \)? Wait, no, \( \triangle ABA \) is wrong. Wait, points \( A, B, C \): \( BA \) and \( BC \) are radii? Wait, \( B \) is the center? Wait, the diagram: \( B \) is the center? Wait, \( BA \) and \( BC \) are radii, so \( \triangle ABC \) is isoceles? Wait, no, the angle at \( B \) is \( 62^\circ \)? Wait, no, the angle between \( BA \) and \( BC \)? Wait, no, the angle given is \( 62^\circ \) between \( BA \) and... Wait, the arc \( AC \): the central angle? Wait, no, wait, \( BC \) is a diameter? Wait, no, the diagram: \( C \) is on the circle, \( B \) is a point, \( A \) is on the circle. Wait, maybe \( B \) is the center? Wait, if \( B \) is the center, then \( BA \) and \( BC \) are radii. Then the angle \( \angle ABC = 62^\circ \)? No, wait, the arc \( AC \): the measure of a minor arc is equal to the measure of its central angle. Wait, but if \( BC \) is a diameter, then the straight line would be \( 180^\circ \). Wait, maybe the angle at \( B \) between \( BA \) and the other side is \( 62^\circ \), so the central angle for arc \( AC \) is \( 180^\circ - 2 \times 62^\circ \)? No, wait, no. Wait, maybe \( \triangle ABB \) is not. Wait, let's re-examine. The diagram: points \( A, B, C \) on the circle? No, \( B \) is the center. So \( BA \) and \( BC \) are radii. The angle between \( BA \) and \( BC \): wait, the angle given is \( 62^\circ \) between \( BA \) and... Wait, maybe the triangle \( \triangle BAA \) is not. Wait, the correct approach: if \( B \) is the center, then the central angle for arc \( AC \) is \( 180^\circ - 2 \times 62^\circ \)? No, wait, no. Wait, the angle at \( B \) is \( 62^\circ \), so the inscribed angle? No, wait, the problem is about the minor arc \( AC \). Wait, maybe \( BC \) is a diameter, so the straight line \( BC \) is \( 180^\circ \). Then the angle between \( BA \) and \( BC \) is \( 62^\circ \), so the angle between \( BA \) and the other side (arc \( AC \)): wait, no. Wait, the measure of arc \( AC \) is equal to the central angle. Wait, if \( B \) is the center, then \( \angle ABC \) is the central angle. Wait, no, the diagram shows \( A \), \( B \), \( C \) with \( B \) at the bottom, \( C \) at the top, \( A \) on the left. So \( BC \) is a vertical line, \( BA \) is a line from \( B \) to \( A \) with angle \( 62^\circ \) between \( BA \) and \( BC \). So the central angle for arc \( AC \) would be \( 180^\circ - 2 \times 62^\circ \)? No, wait, no. Wait, the triangle \( \triangle BAC \): if \( BA = BC \) (radii), then it's isoceles. But the angle at \( B \) is \( 62^\circ \), so the base angles are \( (180 - 62)/2 = 59^\circ \), but that's not relevant. Wait, no, the arc \( AC \): the measure of a minor arc is equal to the central angle. Wait, maybe \( BC \) is a diameter, so the straight angle is \( 180^\circ \). Then the angle between \( BA \) and \( BC \) is \( 62^\circ \), so the angle between \( BA \) and the other side (arc \( AC \)): wait, no. Wait, the correct formula: the measure of arc \( AC \) is \( 2 \times \) the inscribed angle, but no, if \( B \) is the center, then the central angle is equal to the arc measure. Wait, maybe I made a mistake. Wait, the options are 62, 31, 236, 124. Let's think again. If \( BC \) is a diameter, then the arc \( BC \) is \( 180^\circ \). The angle at \( B \) between \( BA \) and \( BC \) is \( 62^\circ \), so the angle between \( BA \) and the other side (arc \( AC \)): wait, no. Wait, the arc \( AC \) is the minor arc, so its measure is \( 180^\circ - 2 \times 62^\circ \)? No, that would be \( 56^\circ \), not an option. Wait, no, maybe the angle given is the inscribed angle. Wait, no, the diagram: \( B \) is on the circle? Wait, maybe \( B \) is on the circle, so \( AC \) is a chord, and \( B \) is a point on the circle. Then the inscribed angle over arc \( AC \) is \( 62^\circ \), so the arc \( AC \) would be \( 2 \times 62^\circ = 124^\circ \). Ah, that makes sense. So if \( B \) is on the circle, then \( \angle ABC \) is an inscribed angle, and the measure of arc \( AC \) is twice the inscribed angle. So \( 2 \times 62^\circ = 124^\circ \). So the answer is D.

Step1: Determine angle type

\( \angle ABC \) is an inscribed angle (since \( B \) is on the circle) subtended by arc \( \widehat{AC} \).

Step2: Apply inscribed angle theorem

The measure of an inscribed angle is half the measure of its subtended arc. So, \( \text{Measure of arc } \widehat{AC} = 2 \times \text{Measure of } \angle ABC \).

Step3: Calculate arc measure

Given \( \angle ABC = 62^\circ \), then \( \text{Measure of } \widehat{AC} = 2 \times 62^\circ = 124^\circ \).

Answer:

D. 124°