4.4.2 Chords and Arcs

Analyzing chord properties, arc length, arc measure, and the relationships between chords and arcs in circles.

定义

A chord is a line segment whose endpoints both lie on a circle. An arc is a portion of the circumference of a circle. The arc measure is the central angle (in degrees) that subtends the arc, while arc length is the actual distance along the circle measured in linear units. Key relationships include: (1) Equal chords in the same circle or congruent circles subtend equal arcs and equal central angles; (2) A chord's perpendicular distance from the center determines its length—the closer the chord to the center, the longer it is; (3) The perpendicular from the center of a circle to a chord bisects the chord; (4) Congruent chords are equidistant from the center of the circle. Arc measure is distinct from arc length: arc measure is angular (measured in degrees or radians), while arc length is the actual curved distance along the circumference.

核心公式

• \(\text{Arc Length} = r\theta\), where \(r\) is the radius and \(\theta\) is the central angle in radians
• \(\text{Arc Length} = \frac{m}{360°} \times 2\pi r\), where \(m\) is the arc measure in degrees
• \(c = 2r\sin\left(\frac{\theta}{2}\right)\), where \(c\) is the chord length, \(r\) is the radius, and \(\theta\) is the central angle in radians
• \(d = r\cos\left(\frac{\theta}{2}\right)\), where \(d\) is the perpendicular distance from the center to the chord, \(r\) is the radius, and \(\theta\) is the central angle
• \(\text{Arc Measure (in degrees)} = \frac{\text{Arc Length}}{2\pi r} \times 360°\)

易错点

• ⚠️ Confusing arc measure with arc length: Arc measure is the central angle in degrees (or radians), while arc length is the actual distance along the circle. Students often forget to convert between these two or use the wrong formula.
• ⚠️ Incorrectly applying the chord-distance relationship: Students may think that a longer chord is always closer to the center, when in fact the opposite is true—a longer chord is closer to the center, and the shortest chord (approaching a point) is farthest from the center.
• ⚠️ Forgetting that the perpendicular from the center bisects the chord: When solving problems involving the distance from the center to a chord, students often fail to recognize that this perpendicular line creates two right triangles, each with half the chord length as one leg.
• ⚠️ Mixing up inscribed angles with central angles: When dealing with arcs subtended by chords, students may confuse the central angle (which equals the arc measure) with an inscribed angle (which is half the arc measure). This leads to incorrect arc measure calculations.