1. Tangents, Arcs, and Chords
CHAPTER 9
Tangents, Arcs, and Chords

2. SECTION 9-1
Basic Terms

3. CIRCLE
is the set of points in a plane at a given distance from a given point in that plane. The given point is the CENTER of the circle and the given distance is the RADIUS.

4. is a segment whose endpoints lie on a circle.
CHORD
is a segment whose endpoints lie on a circle.

5. is a line that contains a chord.
SECANT
is a line that contains a chord.

6. is a chord that contains the center of a circle.
DIAMETER
is a chord that contains the center of a circle.

7. TANGENT
is a line in the plane of a circle that intersects the circle in exactly one point, called the point of tangency.

8. SPHERE
is the set of all points in space at a distance r (radius) from point O (center)

9. Are circles that have congruent radii
CONGRUENT CIRCLES
Are circles that have congruent radii

10. Are spheres that have congruent radii
CONGRUENT SPHERES
Are spheres that have congruent radii

11. Are circles that lie in the same plane and have the same center
CONCENTRIC CIRCLES
Are circles that lie in the same plane and have the same center

12. Are spheres that have the same center.
CONCENTRIC SPHERES
Are spheres that have the same center.

13. INSCRIBED in a CIRCLE
Occurs when each vertex of a polygon lies on the circle and the circle is CIRCUMSCRIBED about the POLYGON

14. SECTION 9-2
Tangents

15. THEOREM 9 -1
If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.

16. Tangents to a circle from a point are congruent
Corollary
Tangents to a circle from a point are congruent

17. THEOREM 9 -2
If a line in the plane of a circle is perpendicular to the radius at its outer endpoint, then the line is tangent to the circle.

18. CIRCUMSCRIBED about the CIRCLE
Occurs when each side of a polygon is tangent to a circle and the circle is INSCRIBED in the POLYGON

19. a line that is tangent to each of two coplanar circles
COMMON TANGENT
a line that is tangent to each of two coplanar circles

20. COMMON Internal TANGENT
Intersects the segment joining the centers of the circles.

21. COMMON External TANGENT
Does not intersect the segment joining the centers of the circles.

22. TANGENT CIRCLES
are coplanar circles that are tangent to the same line at the same point

23. Arcs and Central Angles
SECTION 9-3
Arcs and Central Angles

24. is an angle with its vertex at the center of the circle.
CENTRAL ANGLE
is an angle with its vertex at the center of the circle.

25. is an unbroken part of a circle.
ARC
is an unbroken part of a circle.

26. is the arc formed by two points on a circle
MINOR ARC
is the arc formed by two points on a circle
*The measure of a minor arc is the measure of its central angle

27. is the remaining arc formed by the remaining points on the circle
MAJOR ARC
is the remaining arc formed by the remaining points on the circle
* The measure is 360° minus the minor arc

28. is an arc formed from the endpoints of a circle’s diameter
SEMICIRCLE
is an arc formed from the endpoints of a circle’s diameter
* The measure is 180°

29. Arcs that have exactly one point in common.
ADJACENT ARCS
Arcs that have exactly one point in common.

30. CONGRUENT ARCS
Arcs in the same circle or in congruent circles that have equal measure.

31. POSTULATE 16
The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs

32. THEOREM 9-3
In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent.

33. SECTION 9-4
Arcs and Chords

34. In the same circle or in congruent circles:
THEOREM 9-4
In the same circle or in congruent circles:
Congruent arcs have congruent chords
Congruent chords have congruent arcs

35. THEOREM 9-5
A diameter that is perpendicular to a chord bisects the chord and its arc

36. In the same circle or in congruent circles:
THEOREM 9-6
In the same circle or in congruent circles:
Chords equally distant from the center (or centers) are congruent
Congruent chords are equally distant from the center (or centers)

37. SECTION 9-5
Inscribed Angles

38. INSCRIBED ANGLE
Is an angle whose vertex is on a circle and whose sides contain chords of the circle.

39. Is the intersection of the sides of an inscribed angle and the circle
INTERCEPTED ARC
Is the intersection of the sides of an inscribed angle and the circle

40. THEOREM 9-7
The measure of an inscribed angle is equal to half the measure of its intercepted arc

41. COROLLARY 1
If two inscribed angles intercept the same arc, then the angles are congruent

42. An angle inscribed in a semicircle is a right angle.
COROLLARY 2
An angle inscribed in a semicircle is a right angle.

43. COROLLARY 3
If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary

44. THEOREM 9-8
The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.

45. SECTION 9-6
Other Angles

46. THEOREM 9-9
the measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs.

47. THEOREM 9-10
the measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs

48. Circles and Lengths of Segments
SECTION 9-7
Circles and Lengths of Segments

49. THEOREM 9-11
When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord

50. THEOREM 9-12
When two secant segments are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment.

51. THEOREM 9-13
When a secant and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment is equal to the square of the tangent segment

52. END