Bellwork

1. Clickers Bellwork • Solve for x 3x2+5x+2=0

2. Bellwork Solution • Solve for x 3x2+5x+2=0

3. Section 10.1 Use Properties of Tangents

4. The Concept • Today we’re going to begin our discussion of properties of circles • We’ve discussed circles somewhat before in Chapter 1, but we’re now going to be adding in a new dimension of terminology that has some very interesting applications

5. Definitions As with all the objects we’ve learned about in Geometry, we must learn the terminology before the properties • Circle • Set of all points in a plane that are equidistant from the center of the circle • Radius • Segment whose endpoints are the center and any point on the circle Radius Center

6. Definitions • Chord • Segment whose endpoints are on the circle • Diameter • Chord that contains the center • Secant • Line that intersects the circle in two points • Tangent • Line that intersects at exactly one point, the point of tangency Point of Tangency Diameter Tangent Line Secant Chord

7. On your own Which is the diameter of the circle below? A B C D G E F H

8. On your own Which is a chord of the circle below? A B C D H E F G

9. On your own Which is a secant of the circle below? A B C D G E H F

10. On your own Which is a tangent of the circle below? G B C D H F E A I

11. On your own Which is a center of the circle below? G B C D H F E A I

12. On your own Which is a radius of the circle below? G B C D H F E A I

13. On your own Which is the point of tangency of the circle below? G B C D H F E A I

14. Intersections It’s important to note the different ways that two circles can intersect • 2 points of intersection • 1 point of intersection • No points of intersection These circles intersect at a shared point of tangency, their called tangent circles Circles that share the same center are called concentric

15. Common Tangent • Common Tangent • Line, ray or segment that is tangent to two coplanar circles • The number of common tangents depends on the arrangement of the circles

16. On your own How many common tangents can be found between the two circles below

17. On your own How many common tangents can be found between the two circles below

18. On your own How many common tangents can be found between the two circles below

19. Theorem Theorem 10.1 In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint of the circle b a c

20. Example Is line AB a tangent? B 5 12 A 13

21. On your own Is Segment AB a tangent to the circle? 8 13 15

22. On your own What is the radius of the circle? r 15 12

23. Theorem Theorem 10.2 Tangent segments from a common external point are congruent

24. On your own Solve for x 15 4x-2

25. Homework • 10.1 • 1-11, 12-27 Mult 3, 35-37

26. Practical Example Two flywheels connected with a belt are the basis for most of the add-ons for an engine. If two flywheels, one with a radius of 10 cm are placed in a manner such that the point of tangency of the flywheel is 24 cm from the center of the other, how far apart are the centers of the flywheels?

27. Most Important Points • Parts of a circle • Points of intersection • Common tangents • Perpendicularity of tangents • Congruency of common tangents