Circles

1. Circles

2. A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. circle A, or A

3. In a plane, the interior of a circle consists of the points that are inside the circle. The exterior of a circle consists of the points that are outside the circle. interior exterior

4. Segments and Lines

5. The segment joining the center to a point on the circle is the radius of the circle. Example: Two circles are congruent if they have the same radius measures.

6. A chord is a segment whose endpoints are points on the circle. Ex:

7. A diameter is a chord that passes through the center of the circle. Ex:

8. The distance across the circle, through its center, is the measure of the diameter of the circle. Ex:

9. The length of a diameter is twice the length of a radius. The length of a radius is half the length of a diameter.

10. A secant is a line that intersects a circle in two points

11. A tangent is a line in the plane of a circle that intersects the circle in exactly one point. The point at which a tangent line intersects the circle to which it is tangent is the point of tangency.

12. In a plane, two circles can intersect in two, one or no points. Coplanar circles that intersect in one point are called tangent circles.

13. Coplanar circles that have a common center are called concentric.

14. A line or segment that is tangent to two coplanar circles is called a common tangent.

15. A common internal tangent intersects the segment that joins the centers of the two circles.

16. A common external tangent does not intersect the segment that joins the centers of the two circles.

17. Theorems • If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. • In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

18. Theorem If two segments from the same exterior point are tangent to a circle, then they are congruent.

19. Angles and Arcs

20. In a plane, an angle whose vertex is the center of a circle is a central angle of the circle. central angle

21. If the measure of a central angle is less than 180˚, then B and C and the points of A in the interior of BAC form a minor arc. minor arc

22. The measure of a minor arc is defined to be the measure of its central angle. x x

23. If the two endpoints of an arc are the endpoints of a diameter, then the arc is a semicircle. semicircle

24. The measure of a semicircle is 180˚. semicircle

25. If the measure of a central angle is greater than 180˚, then B and C and the points of A in the exterior of BAC form a major arc of the circle. major arc

26. The measure of a major arc is defined as the difference between 360˚ and the measure of its associated minor arc. x (360 – x)

27. Arc Addition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

28. Inscribed Angles

29. An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle.

30. Measure of an Inscribed Angle If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.

31. Theorem If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

32. Properties of Inscribed Polygons If all the vertices of a polygon lie on a circle, the polygon is inscribed in the circle and the circle is circumscribed about the polygon.

33. Theorems About Inscribed Polygons If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.

34. Theorems About Inscribed Polygons Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.

35. A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

36. Other Angle Relationships in Circles

37. Theorem about Chords of Circles In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

38. Theorem about Chords of Circles If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

39. Theorem about Chords of Circles If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

40. Theorem about Chords of Circles In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

41. Theorem If a tangent and a secant intersect at a point on a circle, then the measure of each angle is one half the measure of its intercepted arc.

42. If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

45. If a tangent and a secant intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

47. If two tangents intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

49. If two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.