Question

1. which of the following is a major central angle?

Explanation:

Step1: Recall central angle definition

A central angle is an angle whose vertex is at the center of a circle, and its sides are radii of the circle. A major central angle is the larger of the two central angles formed by two radii (the other being the minor central angle), and it measures more than \(180^\circ\) (or \(\pi\) radians) but less than \(360^\circ\) (or \(2\pi\) radians). A minor central angle measures less than \(180^\circ\).

Step2: Analyze each diagram

- **First diagram**: The two radii (from center to \(A\) and center to \(E\)) form a straight line? Wait, no, the angle between them: if we consider the circle, the arc opposite the gray angle (the major arc) would correspond to the major central angle. Wait, actually, the central angle in the first diagram: the gray angle and the other angle (around the rest of the circle). Let's see the number of points. The circle has points \(A, B, C, D, E, F, G, H\) (8 points, so each arc between adjacent points is \(360^\circ/8 = 45^\circ\)). In the first diagram, the radii are to \(A\) and \(E\). The number of arcs between \(A\) and \(E\) going the short way: from \(A\) to \(E\) through \(H, G, F\)? Wait, no, \(A\) to \(E\): let's count the steps. \(A\) to \(B\) is 1, \(B\) to \(C\) 2, \(C\) to \(D\) 3, \(D\) to \(E\) 4? Wait, no, maybe the order is \(A, H, G, F, E, D, C, B\)? Wait, no, the circle is labeled clockwise or counter-clockwise? Let's assume counter-clockwise: \(A, B, C, D, E, F, G, H\) back to \(A\). So from \(A\) to \(E\): \(A\) to \(B\) (1), \(B\) to \(C\) (2), \(C\) to \(D\) (3), \(D\) to \(E\) (4) arcs? Wait, no, \(A\) to \(E\) directly: the angle between \(A\) and \(E\) (radii) – if \(A\) and \(E\) are opposite? Wait, 8 points, so opposite points would be 4 apart. \(A\) to \(E\) is 4 steps (each step \(45^\circ\)), so \(4 \times 45^\circ = 180^\circ\)? Wait, no, 8 points: \(A\) (0), \(B\) (45), \(C\) (90), \(D\) (135), \(E\) (180), \(F\) (225), \(G\) (275), \(H\) (315). Oh! So \(A\) is at \(0^\circ\), \(E\) is at \(180^\circ\). Wait, but the angle in the first diagram: the gray angle is between \(A\) and \(E\)? Wait, no, the radii are to \(A\) and \(E\), so the angle between them is \(180^\circ\)? But a major central angle should be more than \(180^\circ\). Wait, maybe I miscounted the points. Wait, maybe the first diagram: the two radii are to \(A\) and \(E\), but the other arc (the major arc) is from \(A\) to \(E\) going the long way (through \(B, C, D\) or \(H, G, F\))? Wait, no, if \(A\) is at \(0^\circ\) and \(E\) is at \(180^\circ\), then the two arcs between \(A\) and \(E\) are both \(180^\circ\), so it's a straight angle. But the other diagrams:

- **Second diagram**: Radii to \(B\) and \(D\). \(B\) is at \(45^\circ\), \(D\) is at \(135^\circ\). The angle between them is \(135^\circ - 45^\circ = 90^\circ\) (minor angle), so the major angle would be \(360^\circ - 90^\circ = 270^\circ\), but the gray angle here is the minor one (90°), so the central angle shown is minor.

- **Third diagram**: Radii to \(G\) and... let's see, the other radius: \(G\) is at \(270^\circ\) (if \(A\) is \(0^\circ\), \(H\) \(315^\circ\), \(G\) \(270^\circ\), \(F\) \(225^\circ\), \(E\) \(180^\circ\), \(D\) \(135^\circ\), \(C\) \(90^\circ\), \(B\) \(45^\circ\)). Wait, the third diagram: radii to \(G\) and... maybe \(E\)? No, the blue radii: one to \(G\) (270°) and one to, say, \(E\) (180°)? Wait, no, the angle between them: \(270^\circ - 180^\circ = 90^\circ\) (minor angle), so the central angle shown is minor.

Wait, going back to the first diagram: maybe the points are arranged differently. Wait, the first diagram: the circle has points \(A, B, C, D, E, F, G, H\) (8 points). The two radii are to \(A\) and \(E\), but the gray angle is the smaller one? No, wait, maybe the first diagram: the central angle (gray) is the minor angle, and the major angle is the other one. Wait, no, the question is "which of the following is a major central angle" – the central angle itself (the angle at the center) that is major. So a major central angle is the angle with measure greater than \(180^\circ\). So in the first diagram, if we consider the two radii, the angle that is the major one: let's see, the number of arcs between the two radii (the sides of the central angle) for the major angle. Let's count the number of intervals between the points on the major arc.

In the first diagram: the two radii are to \(A\) and \(E\). The major arc between \(A\) and \(E\) would pass through \(B, C, D\) (or \(H, G, F\))? Wait, no, if \(A\) is connected to \(E\) with a blue line (diameter?), but the gray angle is on the other side. Wait, maybe the first diagram: the central angle (gray) is actually the minor angle, but the major central angle is the one that's more than \(180^\circ\). Wait, no, the central angle is the angle at the center. So if the two radii form an angle, the major central angle is the larger of the two possible angles (the other being minor). So in the first diagram, the two radii (from center to \(A\) and center to \(E\)): if we look at the circle, the arc from \(A\) to \(E\) going the long way (through \(B, C, D\)) has length corresponding to a central angle greater than \(180^\circ\), and the arc going the short way (through \(H, G, F\)) is less than \(180^\circ\)? Wait, no, with 8 points, each arc between adjacent points is \(45^\circ\). So from \(A\) to \(E\) through \(B, C, D\): that's 4 arcs? Wait, \(A\) to \(B\) (45°), \(B\) to \(C\) (45°), \(C\) to \(D\) (45°), \(D\) to \(E\) (45°): total \(4 \times 45^\circ = 180^\circ\). Through \(H, G, F\): \(A\) to \(H\) (45°), \(H\) to \(G\) (45°), \(G\) to \(F\) (45°), \(F\) to \(E\) (45°): also \(180^\circ\). So that's a straight angle. But maybe the first diagram is actually a central angle where the gray angle is the minor one, and the major one is the rest. Wait, no, maybe I made a mistake. Let's re-express:

A central angle is defined by two radii. The measure of the central angle is equal to the measure of its intercepted arc. So a major central angle intercepts a major arc (an arc that is more than a semicircle, i.e., more than \(180^\circ\)).

- **First diagram**: The intercepted arc for the gray angle? Wait, no, the central angle is the angle at the center. If the two radii are to \(A\) and \(E\), and the arc from \(A\) to \(E\) going the long way (through \(B, C, D\)) is, say, 4 arcs (each 45°), so 4*45=180? No, 8 points: each arc between adjacent points is 360/8=45 degrees. So from \(A\) to \(E\) through \(B, C, D\): that's 4 intervals (A to B, B to C, C to D, D to E), so 4*45=180 degrees. Through \(H, G, F\): also 4 intervals (A to H, H to G, G to F, F to E), 4*45=180 degrees. So that's a semicircle. But maybe the first diagram's central angle is actually the one that's more than 180? Wait, no, maybe the points are not 8? Wait, maybe the first diagram has the two radii forming an angle that is more than 180. Wait, let's look at the other diagrams:

- Second diagram: radii to B and D. B is at, say, 30 degrees, D at 120 degrees: angle between them is 90 degrees (minor), so major angle is 270, but the gray angle is 90 (minor).

- Third diagram: radii to G and, say, E. G is at 270, E at 180: angle between them is 90 degrees (minor), major angle 270.

- First diagram: radii to A and E. If A is at 0, E is at 180, but the other arc (the major arc) would be from A to E through B, C, D (or H, G, F) but that's 180 degrees. Wait, maybe the first diagram's central angle is the one that's actually more than 180. Wait, maybe I miscounted the number of points. Let's assume the circle has 8 points, so each arc is 45 degrees. In the first diagram, the two radii are to A and E, but the angle between them (the gray angle) is the minor angle, and the major angle is the rest. Wait, no, the central angle is the angle at the center. So if the gray angle is the minor one, then the major central angle is the other angle (360 - minor angle). So in the first diagram, if the minor angle is, say, 180? No, that can't be. Wait, maybe the first diagram: the two radii are to A and E, but the arc between A and E that's not the gray arc is the major arc, so the central angle for that major arc is the major central angle. So the central angle in the first diagram (the one corresponding to the major arc) is more than 180 degrees. The other diagrams have central angles less than 180 (90 degrees). So the first diagram's central angle is the major one.

Answer:

The first diagram (the top one) is a major central angle.