Math 9 8.6 & 8.7 Circle Properties 8.6 #1 AND #2 TANGENTS AND CHORDS Property #1 Tangent Line A line that touches a circle only once is called a line. Tangent lines always meet the radius of a circle at degrees, forming a angle. Ex. 1 Find the missing values given a line tangent to each circle: a) b)
Property #2 - Chords Perpendicular line segments from the centre of a circle to a chord cut the chord into two parts; that is, they the chord. When OCB = OCA =, then AC =. Ex. 2 Find the missing values: a) b) c) 8.7 CIRCLE PROPERTIES #3-5 INSCRIBED & CENTRAL ANGLES Property #3 Inscribed Angles A section of the circumference of a circle is an. The shorter arc AB is the arc. The longer arc AB is the arc.
The angle formed by joining the endpoints of an arc to the centre of the circle is the angle. In the diagram above, this is. If we take the same endpoints of an arc and form an angle with any point on the circumference it is called an angle. This is in the diagram above. The central angle and inscribed angles are by the minor arc AB. In a circle, all inscribed angles subtended by the same arc are. Subtended definition and applet: http://www.mathopenref.com/subtend.html PTQ = = Ex. 3 Find the missing angles: a) b) Property #4 Central Angle Theorem In a circle, the central angle subtended by an arc is the measure of an inscribed angle subtended by the same arc. POQ =, or PRQ =
Ex. 4 Find the value of the missing angles. a) b) c) Property #5 Angles in Semicircle A special case of the central angle theorem occurs when the central angle is, also forming the of the circle. Accordingly, the inscribed angle is half the central angle. All inscribed angles subtended by a semicircle are angles. AFB = AGB = AHB =. Ex. 5 Find the values of the missing angles: a) b)
8.6 (#1-10) & 8.7(#11-27) Assignment Circle Properties Find the missing values for each diagram. (#1-4 assume tangent lines) 1. 2. 3. 4. 5. 6. 8 7. 8. 9.
10. 11. 12. 13. 14. 15. 70 50 16. 17. 18.
19. 20. 21. 22. 23. 24. 25. Find the radius 26. Diameter is 36. 27. x 10
8.6 & 8.7 CIRCLE PROPERTIES INQUIRY 1. Tangent http://www.mathopenref.com/tangentline.html What is the angle measurement of a tangent to the radius of a circle? What can you say about the angle measurement of all intersecting tangents and radii? Tangent-Radius Property (p.385) 2. Chords http://www.mathopenref.com/chord.html Click the Show right bisector box. Adjust points P and Q. a) What angle does the chord make with the radius? b) How is a chord divided when it is intersected by the radius? c) What unique part of the circle does the bisector of the chord pass through? Chord Properties (p.393) 3. Inscribed Angles http://www.mathopenref.com/circleinscribed.html Set the angle to 80⁰. What is the angle of any angle inscribed by the same arc? a) For 50⁰ b) For 115⁰ c) For 145⁰ What can you say about all angles subtended by the same arc? Inscribed Angles Property (p.406)
4. Central Angle Theorem http://www.mathopenref.com/arccentralangletheorem.html A. Set the central angle to 100 degrees. What is the measure of the angle on the circumference subtended by the same arc? What is the measure of an angle subtended by the same arc when the: i) Central angle is 120⁰ ii) Central angle is 150⁰ iii) Central angle is 110⁰ What is the relationship between a central angle and an angle subtended by the same arc? Central Angle Theorem (p.406) B. Set the central angle to 180 degrees. What is the measure of the angle on the circumference subtended by the same arc? What is the relationship between the diameter and any angle subtended by the diameter? Angles in a Semicircle Property (p.406)