Tangents to Circles

1. Tangents to Circles Section 10.1

2. Essential Questions • How do I identify segments and lines related to circles? • How do I use properties of a tangent to a circle?

3. Definitions • A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. • Radius – the distance from the center to a point on the circle • Congruent circles – circles that have the same radius. • Diameter – the distance across the circle through its center

4. Diagram of Important Terms center

5. Definition • Chord – a segment whose endpoints are points on the circle.

6. Definition • Secant – a line that intersects a circle in two points.

7. Definition • Tangent – a line in the plane of a circle that intersects the circle in exactly one point.

8. Example 1 • Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius. tangent diameter chord radius

9. Definition • Tangent circles – coplanar circles that intersect in one point

10. Definition • Concentric circles – coplanar circles that have the same center.

11. Definitions • Common tangent – a line or segment that is tangent to two coplanar circles • Common internal tangent – intersects the segment that joins the centers of the two circles • Common external tangent – does not intersect the segment that joins the centers of the two circles

12. Example 2 • Tell whether the common tangents are internal or external. a. b. common internal tangents common external tangents

13. More definitions • Interior of a circle – consists of the points that are inside the circle • Exterior of a circle – consists of the points that are outside the circle

14. point of tangency Definition • Point of tangency – the point at which a tangent line intersects the circle to which it is tangent

15. Perpendicular Tangent Theorem • If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

16. Perpendicular Tangent Converse • 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.

17. central angle Definition • Central angle – an angle whose vertex is the center of a circle.

18. Definitions • Minor arc – Part of a circle that measures less than 180° • Major arc – Part of a circle that measures between 180° and 360°. • Semicircle – An arc whose endpoints are the endpoints of a diameter of the circle. Note : major arcs and semicircles are named with three points and minor arcs are named with two points

19. Diagram of Arcs

20. Definitions • Measure of a minor arc – the measure of its central angle • Measure of a major arc – the difference between 360° and the measure of its associated minor arc.

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

22. Definition • Congruent arcs – two arcs of the same circle or of congruent circles that have the same measure

23. Arcs and Chords Theorem • In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

24. Perpendicular Diameter Theorem • If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

25. Perpendicular Diameter Converse • If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

26. Congruent Chords Theorem • In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

27. Example 3 Use the converse of the Pythagorean Theorem to see if the triangle is right. 112 + 432 ? 452 121 + 1849 ? 2025 1970  2025

28. Congruent Tangent Segments Theorem • If two segments from the same exterior point are tangent to a circle, then they are congruent.

29. Example 4

30. Example 1 • Find the measure of each arc. 70° 360° - 70° = 290° 180°

31. Example 2 • Find the measures of the red arcs. Are the arcs congruent?

32. Example 3 • Find the measures of the red arcs. Are the arcs congruent?

33. Example 4

34. intercepted arc inscribed angle Definitions • Inscribed angle – an angle whose vertex is on a circle and whose sides contain chords of the circle • Intercepted arc – the arc that lies in the interior of an inscribed angle and has endpoints on the angle

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

36. Example 1 • Find the measure of the blue arc or angle. a. b.

37. Congruent Inscribed Angles Theorem • If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

38. Example 2

39. Definitions • Inscribed polygon – a polygon whose vertices all lie on a circle. • Circumscribed circle – A circle with an inscribed polygon. The polygon is an inscribed polygon and the circle is a circumscribed circle.

40. Inscribed Right Triangle Theorem • If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. 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.

41. Inscribed Quadrilateral Theorem • A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

42. Example 3 • Find the value of each variable. b. a.

43. Tangent-Chord Theorem • If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.

44. Example 1

45. Try This!

46. Example 2

47. Interior Intersection Theorem • 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.

48. Exterior Intersection Theorem • If a tangent and a secant, two tangents, or 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.

49. Diagrams for Exterior Intersection Theorem

50. Example 3 • Find the value of x.