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Chapter 10 CIRCLES Ms. Watson Geometry Banneker Academic High School GOAL #1 Identify segments and lines related to circles. Who’s N’ The Circle Fam? VOCABULARY DEFINITION CIRCLE -the set of all points in a plane that are equidistant from a given point called the center of a circle RADIUS -the distance from the center to a point on the circle DIAMETER -the distance across the circle, through its center (the diameter is 2x the radius) radius diameter Who’s N’ The Circle Fam? VOCABULARY DEFINITION -a segment whose endpoints are points on the circle PS and PR are chords CHORD -a chord that passes through the center of the circle DIAMETER -a line that intersects a circle in two points SECANT TANGENT -a line in the plane of a circle that intersects the circle in exactly one point secant R Q tangent P S Who’s N’ The Circle Fam? Common External Tangent Common Tangents A line or line segment that is tangent to two circles in the same plane is called a common tangent. 2 Types Common Internal Tangent GOAL #2 Use properties of tangent to a circle. Theorem 10.1 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. What do we know about right triangles??? How can we use what we know to solve the length of sides of a triangle??? Theorem 10.3 If two segments from the same exterior point are tangent to a circle, then they are congruent (equal). 10.2 Arcs & Chords GOAL #1 Use properties of arcs of circles. Who’s N’ The Circle Fam? VOCABULARY DEFINITION -an angle whose vertex is the center of a circle CENTRAL ANGLE MINOR ARC MAJOR ARC -part of a circle that measures less than 180° SEMICIRCLE -an arc whose endpoints are the end points of a diameter of the circle -part of a circle that measures between 180° and 360° A Minor Arc C K Major Arc B Name That Arc! Arcs are named by their endpoints. MINOR ARCS MAJOR ARCS & SEMICIRCLES -named by their endpoints -the minor arc associated with AKC is AC -named by their endpoints and by a point on the arc -the major arc associated with AKC is ABC A Minor Arc C K Major Arc B Measure That Arc! MINOR ARCS MAJOR ARCS & SEMICIRCLES -the same as the measure of its central angle -the difference between 360° and the measure of its associated minor arc - 60 ° - 360° - 60 ° = 300 ° A 60 ° 300 ° Major Arc Minor Arc 60 ° C K B Arc Addition Postulate Discovery Step 1: Draw a circle. Step 2: Place three points on the circle named A, B, and C. Discover that… The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. mABC = mAB + mBC Arc for Thought • Two arcs of the same circle or of congruent circles are congruent arcs if they have the same measure. So, two minor arcs of the same circle or of congruent circles are congruent if their central angles are congruent. Which minor arcs are congruent? Why are they congruent? GOAL #2 Use properties of chords of circles. Theorem 10.4 A C B Theorem 10.5 Theorem 10.6 If one chord is a perpendicular bisector of another chord, the first chord is a diameter. J M K L Theorem 10.7 Hands on Activity 10.3 Inscribed Angles GOAL #1 Use inscribed angles to solve problems. Who’s N’ The Circle Fam? Inscribed Angle -an angle whose vertex is on a circle and whose sides contain chords of the circle Intercepted Arc -an arc that lies in the interior or an inscribed angle and has endpoints on the angle Theorem 10.8 Measure of an Inscribed Angle If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc. 1 m ABC = mAC A C 2 B Theorem 10.9 If two inscribed angles of a circle intercept the same are, then the angles are congruent. A C B C D D GOAL #2 Use properties of inscribed polugons. Who’s N’ The Circle Fam? If all of the vertices of a polygon lie on a circle, the polygon is INSCRIBED in the circle and the circle is CIRCUMSCRIBED about the polygon. Theorem 10.10 If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Also the angle opposite the diameter A is a right angle. B B is a right angle if and only if AC is a diameter of the circle. C Theorem 10.11 A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. E D, E, F, and G lie on the circle if and only if F G D m D + m F = 180 And m E + m G = 180