Answer:-
In geometry and trigonometry, arcs play a crucial role in understanding the properties of circles and angles. Arcs are segments of circles, and they can be classified into two main types: minor arcs and major arcs. This article will focus on the measurement of the minor arc, specifically the arc AC, and delve into the principles and formulas used to calculate its measure.
A minor arc is a portion of a circle that spans less than 180 degrees. It is defined by two points on the circle, called the endpoints of the arc. In this context, arc AC is the minor arc formed between points A and C on the circle.
To comprehend the measurement of a minor arc, we need to first understand central angles. A central angle is an angle formed by two radii of a circle, with its vertex at the center of the circle. The measure of a central angle is equivalent to the measure of the arc it intercepts.
The central angle theorem states that for any given circle, the measure of a central angle is directly proportional to the measure of its intercepted arc. In mathematical terms, if we denote the measure of the central angle as θ (in degrees) and the measure of the intercepted arc as m, then θ = m.
In the context of the minor arc AC, we can measure it using a protractor, provided we have access to the circle and its endpoints A and C. Aligning the protractor's baseline with the line connecting the circle's center to points A and C, we can read the measure of the minor arc AC directly from the protractor's scale.
Alternatively, we can calculate the measure of the minor arc AC using the central angle theorem and the formula for calculating the circumference of a circle. The formula for the circumference of a circle is C = 2πr, where C represents the circumference and r is the radius of the circle.
To find the measure of arc AC in degrees, we can follow these steps:
1. Determine the circumference of the circle using the formula C = 2πr, where r is the circle's radius.
2. Identify the measure of the central angle ACB (θ) in degrees. This can be done through various geometric or trigonometric methods, such as using the sine, cosine, or tangent ratios for right triangles.
3. Apply the central angle theorem, which states that the measure of the central angle (θ) is equal to the measure of the intercepted minor arc (AC).
4. The measure of the minor arc AC can then be obtained by substituting the value of θ into the formula.
Conclusion:
The measure of the minor arc AC can be determined either by directly measuring it using a protractor or by calculating it using the central angle theorem and the formula for the circumference of a circle. Understanding the properties of arcs, central angles, and the relationship between them provides valuable insights into the geometry of circles and contributes to solving various geometric problems involving circular shapes.
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