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Section 6.1 Circles and Related Segments and Angles

Section 6.1 Circles and Related Segments and Angles. A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle. A circle is named by its center point. The symbol for circle is ⊙.

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Section 6.1 Circles and Related Segments and Angles

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  1. Section 6.1Circles and Related Segments and Angles • A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle. • A circle is named by its center point. • The symbol for circle is ⊙. • A radius is a segment that joins the center of the circle to a point on the circle. • All radii of a circle are congruent. • A line segment that joins two points of a circle is a chord. • A diameter of a circle is a chord that contains the center of the circle Section 6.1 Nack

  2. Segments of a Circle • What are the radii of ⊙Q? • QS, QW, QV, QT • What is the diameter? • WT • What is the chord? • SW • Congruent circles are two or more circles that have congruent radii. • Ex. 1 p. 279, 2 p. 280 Section 6.1 Nack

  3. Concentric Circles are coplanar circles that have a common center. Theorem 6.1.1: A radius that is perpendicular to a chord, bisects the chord. An arc is a segment of a circle determined by two points on the circle and all the points in between. ABC is an arc denoted by ABC. A semicircle is the arc determined by a diameter A minor arc is part of a semicircle. A major arc is more than a semicircle but less than a circle Relationships Concerning Circles O Section 6.1 Nack

  4. A central angle of a circle is an angle whose vertex is the center of the circle and whose sides are radii. An intercepted arc is determined by the two points of intersection of the angle with the circle and all points of the arc in the interior of the angle. ROT is a central angle of ⊙O. RT is the intercepted arc. Angles and Arcs Section 6.1 Nack

  5. Angle and Arc Relationships in the Circle • There are 360∘ in a circle. We can measure arcs in degrees. • Postulate 16: (Central Angle Postulate): In a circle, the degree measure of a central angle is equal to the degree measure of the intercepted arc. • Congruent arcs are arcs with equal measure in either a circle or congruent circles. mAOB = mCOD Ex. 3 p. 281 Section 6.1 Nack

  6. Postulate 17: (Arc-Addition Postulate): If B lies between A and C on a circle, then mAB +mBC = mABC An arc is equal to the sum of its parts. Ex. 4 p. 282 • Inscribed angleof a circle whose vertex is a point on the circle and whose sides are chords of the circle. Fig. 6.15RST is the inscribed angle with chords SR and ST as the sides. • Theorem 6.1.2: The measure of an inscribed angle of a circle is one-half the measure of its intercepted arc. Case 1: One chord is a diameter (below) Case 2: The diameter lies inside the inscribed angle p. 283 a Case 3: The diameter lies outside the inscribed angle p. 283 b Proof Case 1: Draw the radius RO. ROS is isoceles with mR = m S. Since ROT is the exterior angle of ROS then mROT = mR + m S = 2 mS Therefore, mS = ½ mROT. Since the measure of the arc is equal to the measure of the angle then by substitution mS = ½ m RT. Section 6.1 Nack

  7. More on Chords and Arcs • In a circle (or in congruent circles), • Theorem 6.1.3: congruent minor arcs have congruent central angles Fig. 6.17p. 284. • Theorem 6.1.4: congruent central angles have congruent arcs. • Theorem 6.15: congruent chords have congruent minor (major) arcs. • Theorem 6.16: congruent arcs have congruent chords. Section 6.1 Nack

  8. Theorem 6.1.7: Chords that are at the same distance from the center of a circle are congruent. Figure 6.18 • Theorem 6.1.8: Congruent chords are located at the same distance from the center of the circle. Figure 6.18 • Theorem 6.1.9: An angle inscribed in a semicircle is a right angle. Figure 6.19 p. 282 • Theorem 6.1.10: If two inscribed angles intercept the same arc, then these angles are congruent. Figure 6.20 Exercises: 10,30 Section 6.1 Nack

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