Circle Chords and Diameters Secants and Tangents Angles Conclusion Section 4.5: Circles Derek W. Hein Math 3130–1: Modern Geometries Southern Utah University Spring 2014 Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definition Propositions Comments Definition of circle A circle is the set of points that are equidistant from a given point. The given point is called the center, and the given distance is the radius. We recall that two distinct points are sufficient to determine a unique line (in neutral geometry). However, it should be clear that two distinct points determine many circles. Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definition Propositions Comments Proposition A However, the following can be proven: Theorem In the Euclidean plane, three distinct (non–collinear) points determine a unique circle. Proof. Existence: Use the intersection of perpendicular bisectors of two lines to find the center; this also determines the radius. Uniqueness: Proceed by way of contradiction ... (The proof is in the text, pp. 157–159.) Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definition Propositions Comments Proposition B We also note that three distinct non–collinear points determine a triangle! Corollary In Euclidean geometry, every triangle can be circumscribed. Note that the center of the circumscribing circle (called the circumcenter of the triangle) is where the perpendicular bisectors of its sides meet. Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definition Propositions Comments Comments Note that this statement was listed earlier (on p. 127) as being equivalent to the Euclidean parallel postulate. Hence, there are triangles (in hyperbolic geometry) that cannot be circumscribed. (More details in Chapter 6...) Also, we denote the circle C with center O and radius r = OP (determined by a point P on C) by C(O, r). Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Propositions Definitions of chord and diameter Definition A chord of a circle is a line segment that joins any two points on a circle. Definition A diameter of a circle is a chord that contains the center of the circle. Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Propositions Proposition C Theorem If AB is a diameter of a circle and CD is any chord of other the same circle that is not a diameter, then AB > CD. Proof. Text, p. 160. (on board) Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Propositions Proposition D Theorem If a diameter of a circle is perpendicular to a chord of the circle, then the diameter bisects the chord. Proof. Text, p. 161. (on board) Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Propositions Proposition E Theorem If a diameter of a circle bisects a chord of the circle (that is not a diameter), then the diameter is perpendicular to the chord. Proof. Assigned homework. Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Propositions Proposition F Theorem The perpendicular bisector of a chord of a circle contains a diameter of the circle. Proof. Unassigned homework. Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Proposition Definitions of secant and tangent Definition A secant of a circle is a line that contains exactly two points of a circle. Definition A tangent of a circle is a line that contains exactly one point of a circle. Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Proposition Proposition G Theorem If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Proof. Text, p. 162. (on board) Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Arcs Arc Measure Propositions Other Ingredients Definition of central angle Definition Any angle θ whose vertex is the center of a circle is called a central angle for the circle. θ Note that the Protractor Postulate gives us the restriction 0◦ ≤ θ ≤ 180◦ . Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Arcs Arc Measure Propositions Other Ingredients Definition of minor arc Definition If A and B are points on C(O, OA) that are not endpoints of the same diameter, then the union of A, B and all points of C(O, OA) that are in the interior of central angle ∠AOB is called a minor arc of C(O, OA). A ← O B Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Arcs Arc Measure Propositions Other Ingredients Definition of major arc Definition If A and B are points on C(O, OA) that are not endpoints of the same diameter, then the union of A, B and all points of C(O, OA) that are in the exterior of central angle ∠AOB is called a major arc of C(O, OA). A → O B Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Arcs Arc Measure Propositions Other Ingredients Definition of semicircle Definition If A and B are points on C(O, OA) that are endpoints of a diameter, then the union of A, B and all points of C(O, OA) ←→ that lie on either half–plane defined by AB is called a semicircle of C(O, OA). A O B Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Arcs Arc Measure Propositions Other Ingredients Definition of arc measure _ We denote the circular arc between points A and B by AB. Definition Let A and B be points on C(O, OA). Then the degree measure _ of AB is _ a) the degree measure of the angle ∠AOB if AB is a minor arc; _ b) 180◦ if AB is a semicircle; _ c) 360◦ minus the degree measure of the angle ∠AOB if AB is a major arc. _ _ Also, mAB means the measure of arc AB. Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Arcs Arc Measure Propositions Other Ingredients Proposition H Theorem If two chords of a circle are congruent, then their corresponding minor arcs have the same measure. Proof. Immediate, from the above definitions. Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Arcs Arc Measure Propositions Other Ingredients Proposition I (The converse of the last one:) Theorem If two minor arcs of a circle are congruent, then so are the corresponding chords. Proof. Homework! Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Arcs Arc Measure Propositions Other Ingredients Proposition J Many properties of angles carry over to arcs (by SMSG Postulate 13). For example: Theorem (Arc Addition Theorem) _ _ _ _ _ B, then mAPB + mBQC = mABC. If APB and BQC are arcs of a circle sharing only the endpoint Proof. Text, pp. 166–167. Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Arcs Arc Measure Propositions Other Ingredients Definition of inscribed angle Definition _ An angle ∠SP T is said to be inscribed in an arc AQB iff _ a) the vertex P of the angle is a point of AQB, b) one ray of the angle contains point A, and c) the other ray contains point B. A S P Q B Math 3130 T Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Arcs Arc Measure Propositions Other Ingredients Definition of intercepted arc Definition _ An angle ∠SP T is said to intercept an arc ARB iff a) both A and B are points of the angle, b) each ray of the angle contains at least one endpoint of _ ARB, and _ c) (excepting A and B) each point on ARB lies in the interior of ∠SP T . A S P R B Math 3130 T Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Arcs Arc Measure Propositions Other Ingredients Proposition K Theorem (Inscribed Angle Theorem) The measure of an angle inscribed in an arc is one–half the measure of its intercepted arc. Proof. Discussion, pp. 168–169. Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Arcs Arc Measure Propositions Other Ingredients Proposition L Corollary An angle inscribed in a semicircle is a right angle. Proof. Assigned homework. Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Definitions Arcs Arc Measure Propositions Other Ingredients Proposition M Corollary Angles inscribed in congruent (or same) arcs are congruent. Proof. Unassigned homework (?) Math 3130 Section 4.5 Circle Chords and Diameters Secants and Tangents Angles Conclusion Homework Your homework assignment is: pp. 176–180 #3, 5, 7, 8, 11; 17, 22 Math 3130 Section 4.5 Bibliography c Roads to Geometry, 3rd edition, Wallace & West, 2004, Pearson Education Inc., ISBN 978–0–13–041396–3 Math 3130 Section 4.5

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