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TOPIC VII

TOPIC VII. CIRCLES. VII: Circles ESSENTIAL CONTENT. Tangent Properties Tangent to a Circle Tangent Segments Chord Properties Chord Central Angles Arcs Perpendicular to a Chord Chord Distance to Center Perpendicular Bisector of a Chord Arc and Angle Properties

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TOPIC VII

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  1. TOPIC VII CIRCLES

  2. VII: CirclesESSENTIAL CONTENT • Tangent Properties • Tangent to a Circle • Tangent Segments • Chord Properties • Chord Central Angles • Arcs • Perpendicular to a Chord • Chord Distance to Center • Perpendicular Bisector of a Chord • Arc and Angle Properties • Inscribed Angles • Inscribed Angles Intercepting Arcs • Angles Inscribed in a Semicircle • Cyclic Quadrilateral • Parallel Lines Intercepted Arcs • Circumference/Diameter Ratio • Arc Length • Applications in Mathematics and the Real-World

  3. Lesson 6.1Tangent Properties TOPIC VII - CIRCLES

  4. Tangentproperties • You will learn: • Relationship between a tangent line to a circle and the radius of the circle • Relationship between two tangent segments to a common point outside the circle

  5. Tangentproperties TangentDefinition A line that intersects a circle at exactly one point called point of tangency F B E Example: EF

  6. Tangentproperties Tangentconjecture A tangent to a circle is perpendicular to the radius draw to the point of tangency T O

  7. Tangentproperties TangentSegmentsConjecture Tangentsegmentsto a circlefrom a pointoutsidethecircle are congruent A G E N AN GN

  8. Tangentproperties TangentSegmentsconjecture •  BOA determines theminorarc, AB A B •  BOA issaidtointercept AB becauseiswithintheangle. • Themeasure of a minorarcisdefined as themeasure of its central angle, so AB = 400 O 400 • Themeasure of a majorarcisthereflexmeasureof  BOA • or 3600 - themeasure of theminorarc • m BCA = 320

  9. Tangentproperties TangentSegmentsconjectures Example: In the figure at right TA and TG are both tangent to the circle N. If the major arc formed by the two tangents measure 2200, find the measure of T A G Solution: The minor arc intercepted by N measures 3600 – 2200 or 1400 N T m  N = 1400 2200  A and  G must be right angles By quadrilateral conjecture the sum of the angles in TANG is 3600 So, m  T + 90 0 + 140 0 + 90 0 = 360 0 means that m  T = 40

  10. Tangentproperties Tangentcircles: Are twocirclesthat are tantgenttothesame line at thesamepoint. They can be: Internallytangent Externallytangent

  11. Tangentproperties Summary The main points of this lesson are that Every line tangent to a circle is perpendicular to the radius at the point of tangency The two tangent segments from a point outside the circle are congruent.

  12. Tangentproperties TangentSegmentsconjectures Exercises: 1. Rays m and n are tangent to circle P. Find mw 2. Rays r and s are tangent to circle Q. Find x m r P Q 1300 w 700 x n n

  13. Tangentproperties TangentSegmentsconjectures Exercises: 4. Line t is tangent to both tangent circles. Find z 3. Ray K is tangent to circle R. Find y r y t 750 z S M 1200 R K

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