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Circle Geometry: Segments, Lines & Tangents

Identify segments and lines related to circles, explore properties of tangents. Learn about radius, diameter, chords, secants, and tangents in circles. Solve problems involving intersection points, circle centers, radii, tangents, and chords. Understand the concepts of tangency, common tangent lines, and internal and external tangents. Apply theorems related to tangents and segments in circles. Practice finding intersection points, common tangents, slopes, and measures of arcs. Explore inscribed angles and central angles in circles for further problem-solving.

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Circle Geometry: Segments, Lines & Tangents

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  1. Lesson 10.1 Today, we are going to… > identify segments and lines related to circles > use properties of tangents to a circle Parts of a Circle

  2. radius diameter C Circle C 2 Diameter = _ radius

  3. Y N BN YX AB A C X B A chord is a segment whose endpoints are on the circle

  4. Y YX AB A C X B A secant is a line that intersects the circle in 2 points

  5. AB XY X Y C B A A tangent is a line that intersects the circle in exactly one point Point of tangency

  6. Common Tangent Lines internal tangents

  7. Common Tangent Lines external tangents

  8. 1. Point of Tangency 2. Internal Tangent 8. External Tangent 4. Chord 7. Secant 3. Radius 5. Center 6. Diameter

  9. Two circles can intersect in 2, 1, or 0 points. Draw 2 circles that have 2 points of intersection

  10. internally tangent circles Draw two circles that have 1 point of intersection

  11. externally tangent circles Draw two circles that have 1 point of intersection

  12. concentric circles Draw two circles that have no point of intersection

  13. 9. What are the center and radius of circle A? Center: Radius = (3, 2) 2

  14. 10. What are the center and radius of circle B? Center: Radius = (7, 2) 2

  15. 11. Identify the intersection of the two circles. (5, 2)

  16. 12. Identify all common tangents of the two circles. y = 4 y = 0 x = 5

  17. ° m Ð ABC = 90 B C A

  18. Theorem 10.1 & 10.2 A line is tangent to a circle if and only if it is perpendicular to the radius from the point of tangency.

  19. 13. Find CA. (CA)2 = 72 + 152 C 7 (CA)2 = 274 CA=16.6 D B 15 What is DA? A 16.6 – 7 = 9.6

  20. 14. Find x. C x 7 x x What is CA? (x+8)2 = x2 + 162 B 8 6 x2 + 16x + 64 = x2 + 256 16 15 16x + 64 = 256 A x = 12

  21. 15. Is AB a tangent? Yes, AB is a tangent How do we test if 3 segments create a right triangle? C 10 7 262 = 242 + 102 26 B 676 = 576 + 100 6 24 15 A

  22. 16. Is AB a tangent? No, AB is not a tangent C 8 7 172 = 122 + 82 17 289 = 144 + 64 B 6 12 289 = 208 15 A

  23. Slope of AC? A C 17. Find the slope of line t. A (3,0) and C (5, -1) t - ½ Slope of line t? 2

  24. A B C A tangent segment lies in the line that is tangent to the circle One endpoint is the point of tangency.

  25. Theorem 10.3 GSP

  26. Theorem 10.3 If 2 segments from the same point outside a circle are tangent to the circle, then they are congruent.

  27. 18. Find x. 7x – 2 = 3x + 8 4x = 10 B x = 2.5 7x - 2 A 3x + 8 C

  28. 19. Find x. x2 + 25 = 50 x2 = 25 B x = 5 x2 + 25 A 50 C

  29. 20. Find x. 130 + 90 + 90 + x = 360 B x = 50 130° A x° C

  30. Lesson 10.2 Arcs and Chords Today, we are going to… > use properties of arcs and chords of circles

  31. C An angle whose vertex is the center of a circle is a central angle. A B

  32. Major Arc ADB C Minor Arc AB Minor Arc - Major Arc D A B

  33. D A 60˚ C B m AB = Measures of Arcs 60° Central angles are equal to their arcs.

  34. E D A B m AED = m ABD = m AD Semicircle C

  35. Find the measures of the arcs. 1. m BD 2. m DE 3. m FC 4. m BFD D C =120º 68˚ =87º 52˚ 87º ? B 100˚ E =152º 53˚ =240º F

  36. E F D C A B AD and EB are diameters. 5. Find x, y, and z. x = 60 x˚ 30˚ y = 60 y˚ z = 120 z˚

  37. Theorem 10.4 GSP

  38. Theorem 10.4 Two arcs are congruent if and only if their chords are congruent.

  39. 6. Find m AB mAB = 122° (2(37) + 48)° B (3x + 11)° (2x + 48)° C A D x = 37 3x + 11 = 2x + 48

  40. Theorem 10.5 & 10.6 GSP

  41. Theorem 10.5 & 10.6 A chord is a diameter if and only if it is a perpendicular bisector of a chord and bisects its arc.

  42. 7. Is AB a diameter? A B No,AB is  to the other chord but does not bisect it.

  43. 8. Is AB a diameter? A 8 8 B No,AB bisects the other chord, but it is not  to it.

  44. Yes,AB is a  bisector of the other chord. 9. Is AB a diameter? A B

  45. Theorem 10.7 GSP

  46. Theorem 10.7 Two chords are congruent if and only if they are equidistant from the center.

  47. ≈ 3.6 B G A C D F x = ≈ 3.6 E 10. Find CG. CG = AB = 12 DE = 12 x 7 72 = x2 + 62 6 ?

  48. Lesson 10.3 Inscribed Angles Today, we are ALSO going to… > use properties of inscribed angles to solve problems

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