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Transcript
Circles Chapter 12 Parts of a Circle A • E• Circle: The set of all points in a plane •B that are a given distance from a given point in that plane. Center: The middle of the circle – a M• diameter circle is named by its center, the symbol of a circle looks like - סּM. •C • D Radius: a segment that has one endpoint at the center and the other endpoint on the circle. The radius is ½ the length of the diameter. ALL RADII ARE CONGRUENT. More Vocabulary A • Chord: a segment that has its •B endpoints on the circle. Diameter: a chord that passes E• through the center of the circle. The diameter is 2 times the radius. M• Circumference: the distance diameter •C • D around the circle. To find the circumference use: C 2r Arcs: the space on the circle between the two points on the circle. Tangent Lines Tangent line: A line that intersects the circle at exactly one point. AB is a tangent line to סּT. Point of Tangency: the point where the circle and the tangent line intersect. •A •P T• Theorem 12-1: If a line is tangent to a circle, then the line is perpendicular to a radius drawn to the point of tangency. AB TP •B Tangent Lines Continued •Q P • • X Theorem 12-3: Two •R segments tangent to a circle from the same point outside of the circle are congruent. PQ QR Central Angles Central Angles: angles whose vertex is the center of the circle. •G D• Theorem 12-4: 37º Within a circle or congruent circles: 37º (1) Congruent central angles have congruent chords. •E F• DF DG (2) Congruent chords have congruent arcs. DF DG (3) Congruent arcs have congruent central angles. GED DEF Chords Theorem 12-5: Within a circle or congruent circles (1) Chords equidistant from the center are congruent. AB CD (2) Congruent chords are equidistant from the center. EP PF •B E • A• P• C• •D •F More About Chords Theorem 12-6: In a circle, a diameter that is perpendicular to a chord bisects the chord and its arc. UV VS •U UR RS Theorem 12-7: In a circle, a diameter that bisects a Q • chord (that is not a diameter) is perpendicular to the chord. QR US Theorem 12-8: In a circle, the perpendicular bisector of a chord contains the center of the circle. • T V • •S •R Inscribed Angles Inscribed Angle: An angle whose vertex is on the circle, and the sides are chords of the circle. C Intercepted Arc: an arc of a circle having endpoints on the sides of an inscribed angle. AB A • D• B • •C Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc. A ● 90° 1 mB mAC 2 Corollaries: 1. Two inscribed angles that intercept the same arc are congruent. 2. An inscribed angle in a semicircle is a right angle. 3. The opposite angles of a quadrilateral inscribed in a circle are 1 supplementary. 4 ●B 45° C● 38° 1 380 ● 1 m2 90 2 3 ● 0 ● 2 ● m1 m3 1800 An angle formed by a tangent line and a chord. The measure of angle formed by a tangent line and a chord is half the measure of the intercepted arc. B ● 1 mC mBDC 2 ● D● 130° 65° ● C Secant Line A secant line is a line that intersects a circle at two points. Theorem 12-11: The measure of an angle formed by two lines that (1) Intersect inside a circle is half the sum of the measure of the intercepted arcs. x° 1 m1 x y 2 1 (2) Intersect outside the circle is half the difference of the measures of the intercepted arcs. 1 m1 2 x y y° ● x° 1 ● ● y° ● Segment Length Theorems 3. (y + z)y = t2 1. a ● b = c ● d t c a d ● b ● 2. (w + x)w = (y + z)y z x w z y ● y Equation of a Circle An equation of a circle with the center (h, k) and radius r is (x – h)2 + (y – k)2 = r2. Example: Center (5, 3) radius 4. (x – 5)2 + (y – 3)2 = 16