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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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 = ?

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

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 = ?

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

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 = ?

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

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

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

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

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

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

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 from above statement

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 from above statement Using Pythagorean theorem…

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 from above statement Using Pythagorean theorem… Diameter = 10

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

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

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 = RQ = 8 which was given 8 Again use Pythagorean theorem… t

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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