Circles Chapter 12

Parts of a Circle A • M B C D E Chord diameter radius Circle: The set of all points in a plane that are a given distance from a given point in that plane. Center: The middle of the circle – a circle is named by its center, the symbol of a circle looks like - סּM. 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 • M B C D E Chord diameter radius Chord: a segment that has its endpoints on the circle. Diameter: a chord that passes through the center of the circle. The diameter is 2 times the radius. Circumference: the distance around the circle. To find the circumference use: 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. is a tangent line to סּT. • T P A B Point of Tangency: the point where the circle and the tangent line intersect. 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.

Tangent Lines Continued • X P Q R Theorem 12-3: Two segments tangent to a circle from the same point outside of the circle are congruent.

Central Angles Central Angles: angles whose vertex is the center of the circle. • D E F G 37º Theorem 12-4: Within a circle or congruent circles: Congruent central angles have congruent chords. Congruent chords have congruent arcs. Congruent arcs have congruent central angles.

Chords • Theorem 12-5: Within a circle or congruent circles Chords equidistant from the center are congruent. Congruent chords are equidistant from the center. • P A E B C F D

More About Chords • Theorem 12-6: Theorem 12-7: Q Theorem 12-8: In a circle, a diameter that is perpendicular to a chord bisects the chord and its arc. • T V U R S Q Theorem 12-7: In a circle, a diameter that bisects a chord (that is not a diameter) is perpendicular to the chord. Theorem 12-8: In a circle, the perpendicular bisector of a chord contains the center of the circle.

Inscribed Angles Inscribed Angle: An angle whose vertex is on the circle, and the sides are chords of the circle. • D A B C Intercepted Arc: an arc of a circle having endpoints on the sides of an inscribed angle. AB

Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc. ● A B C 90° 45° Corollaries: Two inscribed angles that intercept the same arc are congruent. An inscribed angle in a semicircle is a right angle. The opposite angles of a quadrilateral inscribed in a circle are supplementary. 1 38° ● 2 1 2 3 4

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. ● C D B 65° 130°

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 Intersect inside a circle is half the sum of the measure of the intercepted arcs. Intersect outside the circle is half the difference of the measures of the intercepted arcs. x° 1 y° 1 y° ● x°

Segment Length Theorems 3. (y + z)y = t2 1. a ● b = c ● d t y z ● a b c d ● 2. (w + x)w = (y + z)y w x y z ●

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