3 0.25 points Name the major arc and find its measure. B A 75 ^ C D Arc type your answer... Previous The measure of the arc is type your answer... degrees
The Correct Answer and Explanation is:
Arc: ADB
The measure of the arc is: 285 degrees
Explanation
This problem requires us to identify the major arc in the given circle and calculate its measure.
First, it is important to understand the definitions of different types of arcs in a circle. A minor arc is the shorter path along the circle between two points, and its measure is always less than 180 degrees. A major arc is the longer path between the same two points, and its measure is always greater than 180 degrees. A full circle comprises 360 degrees.
In the provided diagram, we have a circle with its center at point C. The points A, B, and D are located on the circumference of the circle. We are given the measure of the central angle ∠ACB, which is 75 degrees. The measure of a minor arc is equal to the measure of its corresponding central angle. In this case, the central angle ∠ACB subtends the minor arc that connects points A and B. Therefore, the measure of the minor arc AB is 75 degrees.
The question asks for the name and measure of the major arc. The major arc is the longer path from point A to point B that does not pass directly between them. To avoid confusion with the minor arc, a major arc is typically named using three letters: the two endpoints and another point that lies on the arc. In this diagram, the point D is on the longer path between A and B, so the major arc is named arc ADB.
To find the measure of the major arc ADB, we subtract the measure of the corresponding minor arc AB from the total degrees in a full circle.
The calculation is performed as follows:
Measure of major arc ADB = (Total degrees in a circle) – (Measure of minor arc AB)
Measure of major arc ADB = 360° – 75°
Measure of major arc ADB = 285°
Thus, the major arc is correctly named arc ADB, and its measure is 285 degrees.
