1. Circles - Introduction Circle – the boundary of a round region in a plane A

2. Circles - Introduction Circle – the boundary of a round region in a plane A - The set of all points in a plane that are given distance from a given point in the plane. P

3. Circles - Introduction Circle – the boundary of a round region in a plane A -The set of all points in a plane that are given distance from a given point in the plane. -These points form a round line around point P… P

4. Circles - Introduction Circle – the boundary of a round region in a plane A -The set of all points in a plane that are given distance from a given point in the plane. -These points form a round line around point P… P The center of this circle is the given point in the middle, point P. Circles are named by their center point. So in this case we have circle P.

5. Circles - Introduction Circle – the boundary of a round region in a plane A -The set of all points in a plane that are given distance from a given point in the plane. -These points form a round line around point P… P The center of this circle is the given point in the middle, point P. Circles are named by their center point. So in this case we have circle P. Radius – a line segment from the center out to the edge of the circle - it measures the distance from the center to any point on the circle O

6. Circles - Introduction Circle – the boundary of a round region in a plane A -The set of all points in a plane that are given distance from a given point in the plane. -These points form a round line around point P… P The center of this circle is the given point in the middle, point P. Circles are named by their center point. So in this case we have circle P. Radius – a line segment from the center out to the edge of the circle - it measures the distance from the center to any point on the circle Diameter – a line segment that has endpoints on the circle and goes through the center O N M

7. Circles - Introduction Circle – the boundary of a round region in a plane A -The set of all points in a plane that are given distance from a given point in the plane. -These points form a round line around point P… P The center of this circle is the given point in the middle, point P. Circles are named by their center point. So in this case we have circle P. Radius – a line segment from the center out to the edge of the circle - it measures the distance from the center to any point on the circle Diameter – a line segment that has endpoints on the circle and goes through the center - two times larger than the radius O N M

8. Circles - Introduction Circle – the boundary of a round region in a plane A -The set of all points in a plane that are given distance from a given point in the plane. -These points form a round line around point P… PO N M Chord – a line segment that joins any two points on the circle. S R

9. Circles - Introduction Circle – the boundary of a round region in a plane A -The set of all points in a plane that are given distance from a given point in the plane. -These points form a round line around point P… PO N M Chord – a line segment that joins any two points on the circle. The interior of the circle are the points contained inside the circle ( blue shading ) The exterior of the circle are the points sitting outside the circle ( gray shading ) S R

10. Circles - Introduction Circle – the boundary of a round region in a plane A -The set of all points in a plane that are given distance from a given point in the plane. -These points form a round line around point P… PO N M Chord – a line segment that joins any two points on the circle. The interior of the circle are the points contained inside the circle ( blue shading ) The exterior of the circle are the points sitting outside the circle ( gray shading ) Multiple radii can be drawn from the center. S RT

11. Circles - Introduction Line Chord Theorem : If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord A P Q MN a

12. Circles - Introduction Line Chord Theorem : If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord It’s referred to as a perpendicular bisector A P Q MN a

13. Circles - Introduction Line Chord Theorem : If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord It’s referred to as a perpendicular bisector The converse is true as well : If a line through the center of a circle bisects a chord, it is perpendicular to that chord. A P Q MN a

14. Circles - Introduction Line Chord Theorem : If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord It’s referred to as a perpendicular bisector It cuts segment MN in half A P Q MN a

15. Circles - Introduction Line Chord Theorem : If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord It’s referred to as a perpendicular bisector It cuts segment MN in half If we draw to radii, we create two right triangles A P Q MN a

16. Circles - Introduction Line Chord Theorem : If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord It’s referred to as a perpendicular bisector It cuts segment MN in half If we draw to radii, we create two right triangles These two triangles are congruent. A P Q MN a

17. Circles - Introduction Line Chord Theorem : If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord It’s referred to as a perpendicular bisector It cuts segment MN in half If we draw to radii, we create two right triangles These two triangles are congruent. A P Q MN a EXAMPLE : Fill in the table PN = 10PM = ?

18. Circles - Introduction Line Chord Theorem : If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord It’s referred to as a perpendicular bisector It cuts segment MN in half If we draw to radii, we create two right triangles These two triangles are congruent. A P Q MN a EXAMPLE : Fill in the table PN = 10PM = 10

19. Circles - Introduction Line Chord Theorem : If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord It’s referred to as a perpendicular bisector It cuts segment MN in half If we draw to radii, we create two right triangles These two triangles are congruent. A P Q MN a EXAMPLE : Fill in the table PN = 10PM = 10 QM = 8QN = ?

20. Circles - Introduction Line Chord Theorem : If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord It’s referred to as a perpendicular bisector It cuts segment MN in half If we draw to radii, we create two right triangles These two triangles are congruent. A P Q MN a EXAMPLE : Fill in the table PN = 10PM = 10 QM = 8QN = 8

21. Circles - Introduction Line Chord Theorem : If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord It’s referred to as a perpendicular bisector It cuts segment MN in half If we draw to radii, we create two right triangles These two triangles are congruent. A P Q MN a EXAMPLE : Fill in the table PN = 10PM = 10 QM = 8QN = 8 If MN = 30QN = ?

22. Circles - Introduction Line Chord Theorem : If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord It’s referred to as a perpendicular bisector It cuts segment MN in half If we draw to radii, we create two right triangles These two triangles are congruent. A P Q MN a EXAMPLE : Fill in the table PN = 10PM = 10 QM = 8QN = 8 If MN = 30QN = 15

23. Circles – Introduction If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord Z C AB

24. Circles – Introduction If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord With this in mind, we can solve problems like this one. Z C AB R S

25. Circles – Introduction If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord With this in mind, we can solve problems like this one. Z C AB R S Solution : First draw in your radius

26. Circles – Introduction If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord With this in mind, we can solve problems like this one. Z C AB R S Solution : First draw in your radius 3

27. Circles – Introduction If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord With this in mind, we can solve problems like this one. Z C AB R S Solution : First draw in your radius 3 4

28. Circles – Introduction If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord With this in mind, we can solve problems like this one. Z C AB R S Solution : First draw in your radius 3 4 - from above statement

29. Circles – Introduction If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord With this in mind, we can solve problems like this one. Z C AB R S Solution : First draw in your radius 3 4 - from above statement Using Pythagorean theorem…

30. Circles – Introduction If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord With this in mind, we can solve problems like this one. Z C AB R S Solution : First draw in your radius 3 4 - from above statement Using Pythagorean theorem… Diameter = 10

31. Circles – Introduction Example # 2 : Diameter of Circle M = 20 Segment RQ = 8 Find MQ Z t M R Q S

32. Circles – Introduction Example # 2 : Diameter of Circle M = 20 Segment RQ = 8 Find MQ Z S M R Q 10 RQ = 8 which was given 8 t Solution : If Diameter of M = 20, then MR = 10

33. Circles – Introduction Example # 2 : Diameter of Circle M = 20 Segment RQ = 8 Find MQ Z S M R Q Solution : If Diameter of M = 20, then MR = 10 10 RQ = 8 which was given 8 Again use Pythagorean theorem… t

34. Circles – Introduction Example # 3 : Circle M = Circle N Line t bisects and is perp. to SR and AB AB = SR SR = 72 MQ = 48 Find ND S M R D N Q B A C t

35. Circles – Introduction Example # 3 : Circle M = Circle N Line t bisects and is perp. to SR and AB AB = SR SR = 72 MQ = 48 Find ND S M R Solution : If circle M = circle N, then the parts of these circles are congruent, which would include radius, diameter, and chords. D N Q B A C t

36. Circles – Introduction Example # 3 : Circle M = Circle N Line t bisects and is perp. to SR and AB AB = SR SR = 72 MQ = 48 Find ND S M R Solution : If circle M = circle N, then the parts of these circles are congruent, which would include radius, diameter, and chords. It was given that AB = SR and SR = 72, so AB = 72. D N Q B A C t

37. Circles – Introduction Example # 3 : Circle M = Circle N Line t bisects and is perp. to SR and AB AB = SR SR = 72 MQ = 48 Find ND S M R Solution : If circle M = circle N, then the parts of these circles are congruent, which would include radius, diameter, and chords. It was given that AB = SR and SR = 72, so AB = 72. If AB = 72, AC and CB = 36. D N Q B A C t

38. Circles – Introduction Example # 3 : Circle M = Circle N Line t bisects and is perp. to SR and AB AB = SR SR = 72 MQ = 48 Find ND S M R Solution : If circle M = circle N, then the parts of these circles are congruent, which would include radius, diameter, and chords. It was given that AB = SR and SR = 72, so AB = 72. If AB = 72, AC and CB = 36. If MQ = 48, then NC = 48. D N Q B A C t

39. Circles – Introduction Example # 3 : Circle M = Circle N Line t bisects and is perp. to SR and AB AB = SR SR = 72 MQ = 48 Find ND S M R Solution : If circle M = circle N, then the parts of these circles are congruent, which would include radius, diameter, and chords. It was given that AB = SR and SR = 72, so AB = 72. If AB = 72, AC and CB = 36. If MQ = 48, then NC = 48. That gives us 2 sides of a right triangle. ( ∆NAC ) D N Q B A C t N A C 48 36

40. Circles – Introduction Example # 3 : Circle M = Circle N Line t bisects and is perp. to SR and AB AB = SR SR = 72 MQ = 48 Find ND S M R Solution : If circle M = circle N, then the parts of these circles are congruent, which would include radius, diameter, and chords. It was given that AB = SR and SR = 72, so AB = 72. If AB = 72, AC and CB = 36. If MQ = 48, then NC = 48. That gives us 2 sides of a right triangle. ( ∆NAC ) If we can find NA, we will know ND, because ND is also a radius of circle N. D N Q B A C t N A C 48 36

41. Circles – Introduction Example # 3 : Circle M = Circle N Line t bisects and is perp. to SR and AB AB = SR SR = 72 MQ = 48 Find ND S M R Solution : If circle M = circle N, then the parts of these circles are congruent, which would include radius, diameter, and chords. It was given that AB = SR and SR = 72, so AB = 72. If AB = 72, AC and CB = 36. If MQ = 48, then NC = 48. That gives us 2 sides of a right triangle. ( ∆NAC ) If we can find NA, we will know ND, because ND is also a radius of circle N. D N Q B A C t N A C 48 36

42. Circles – Introduction Example # 3 : Circle M = Circle N Line t bisects and is perp. to SR and AB AB = SR SR = 72 MQ = 48 Find ND S M R Solution : If circle M = circle N, then the parts of these circles are congruent, which would include radius, diameter, and chords. It was given that AB = SR and SR = 72, so AB = 72. If AB = 72, AC and CB = 36. If MQ = 48, then NC = 48. That gives us 2 sides of a right triangle. ( ∆NAC ) If we can find NA, we will know ND, because ND is also a radius of circle N. If NA = 60, then ND = 60 D N Q B A C t N A C 48 36

43. Circles – Introduction Chords that are the same distance from the center of a circle have equal length. M

44. Circles – Introduction Chords that are the same distance from the center of a circle have equal length. This also applies to congruent circles. If circle M = circle N, and if the red distances are equal, then CD = RS M N D C S R

45. Circles – Introduction Chords that are the same distance from the center of a circle have equal length. This also applies to congruent circles. If circle M = circle N, and if the red distances are equal, then CD = RS EXAMPLE : N D C A B X Y XY bisects and is perpendicular to AB and CD. AB = 24 and NB = 20, find XY.

46. Circles – Introduction Chords that are the same distance from the center of a circle have equal length. This also applies to congruent circles. If circle M = circle N, and if the red distances are equal, then CD = RS EXAMPLE : N D C A B X Y XY bisects and is perpendicular to AB and CD. AB = 24 and NB = 20, find XY. 12

47. Circles – Introduction Chords that are the same distance from the center of a circle have equal length. This also applies to congruent circles. If circle M = circle N, and if the red distances are equal, then CD = RS EXAMPLE : N D C A B X Y XY bisects and is perpendicular to AB and CD. AB = 24 and NB = 20, find XY. 12 20

48. Circles – Introduction Chords that are the same distance from the center of a circle have equal length. This also applies to congruent circles. If circle M = circle N, and if the red distances are equal, then CD = RS EXAMPLE : N D C A B X Y XY bisects and is perpendicular to AB and CD. AB = 24 and NB = 20, find XY. 12 20 16

49. Circles – Introduction Chords that are the same distance from the center of a circle have equal length. This also applies to congruent circles. If circle M = circle N, and if the red distances are equal, then CD = RS EXAMPLE : N D C A B X Y XY bisects and is perpendicular to AB and CD. AB = 24 and NB = 20, find XY. 12 20