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Name ________________________ Worksheet 6.1 Circles Before embarking on our study of the circle, we define the following: Circle: the set of all points equidistant from a common point (the center of the circle) Radius: a line segment connecting the center point to a point on the circle Chord: a line segment connecting two points on the circle We also present a postulate regarding radii of a circle: all radii of a circle are congruent Using these definitions and postulate, prove the following theorem in paragraph form: Theorem: If a radius is perpendicular to a chord of a circle, then it bisects the chord. Q: Is the converse of this theorem true? ________ Here is an additional definition regarding circles: Tangent: a line that intersects a circle only once (a tangent is always perpendicular to a radius drawn to the point of tangency) Using this definition of a tangent, prove the following, also in paragraph form: Theorem: If two tangent segments share a common endpoint outside the circle, then they are congruent. Measures of Arcs By angle: An arc can be measured by its central angle, for example, a 70º arc. By arc length: The length of an arc can be calculated simply by understanding it as a fraction of a circumference. Arc length = _______ 2R Exercises: 1. Find the arc length. 6 82º 82º 2. This circle has radius 10 and arc AB has length 15. Find the measure of arc AB in degrees. A AB=15 B A Xº R Xo B Inscribed Angle Theorem 2X An inscribed angle is half the size of the central angle to the same arc. X Inscribed Angle and Its Intercepted Arc Our work with the circle leads us to encounter and work with the inscribed angle quite often. While we have been exposed to the relationship that exists between this angle and its intercepted arc, we have yet to fully understand why this relationship exists. Let us now take the time to prove the following theorem: Theorem: If an angle is an inscribed angle, then it has a measure equal to one-half that of its intercepted arc. Since an intercepted arc may present itself in many forms (scenarios), we must consider proving the theorem in all these scenarios. Scenario #1: Given: AP is a diameter Prove algebraically (with proper discussion) that mAPB 1 m AB 2 (Hint: draw radius BO) P O B A Scenario #2: Prove algebraically (with proper discussion) that mAPB 1 m AB 2 (Hint: draw diameter through P & O) P A O B Scenario #3: Prove algebraically (with proper discussion) that mAPB 1 m AB 2 (Hint: draw diameter through P & O) P O A B Exercises: 3. Find the arcs and angles labeled (note the center is not part of the figure in the circle). 100º w 58º y z v x 120º 4. This quadrilateral circumscribes the circle. Find the missing side. 21 32 25 Useful Corollaries to the Inscribed Angle Theorem An inscribed triangle with one Opposite angles in an inscribed An inscribed "butterfly" like side a diameter is a right triangle. quadrilateral are this is made up of similar supplementary. triangles. 107º 92º 88º 73º Exercises: 5. Label all missing angles and arcs. 68º 65º 72º 6. Find angles X and Y. Then find length A. 80º 3 2 Y 5 A 36º X 7. Find angle X (hint: draw in AB). X 47º A 149º B 8. Explain why this is a true theorem. If two chords are equal then they have equal arcs and central angles and vice versa. 8. 9. 9. The circles have radii of 5 and 11. AB and CX are tangents. Find AX , XB , and XC . B X A C 10. One piece of a locomotive gear recovered in a train accident must be used to make a replacement. AB is carefully measured to be 81.9 cm. CD, the perpendicular bisector of AB, is measured to be 24.1 cm. Find the radius of the original gear. Maintain enough accuracy to be sure your final value is accurate to 0.1 cm. C A D B 11. Find the missing side of the circumscribed quadrilateral. 64 83 57 12. Two tangent circles of radius 7 and 17 have a common tangent AB. Find the length of AB. A B 13. Find X. (Hint: You will need to draw in one chord). 17º X 114º 14. Find all missing arcs and angles. 110º 24º 156º