Newport Math Club Circles. What is a Circle? Original Content © 2008 Newport Math Club. Images adapted from mathwarehouse.com

Newport Math ClubNewport Math Club

Circles

What is a Circle?What is a Circle?

Original Content © 2008 Newport Math Club. Images adapted from mathwarehouse.com.

Important VocabsImportant Vocabs

►CenterCenter►Radius (r)Radius (r)►Diameter (d=2r)Diameter (d=2r)►Circumference Circumference

((2πr2πr))►Area (Area (ππr²)r²)►AngleAngle

Degree (360Degree (360°°per per circle)circle)

Radian (2Radian (2π per circleπ per circle))Original Content © 2008 Newport Math Club. Images adapted from mathwarehouse.com.

More VocabsMore Vocabs

►ArcArc►SectorSector►TangentTangent►SecantSecant►ChordChord►SegmentSegment


Arc, Sector, SegmentArc, Sector, Segment

► Arc: any unbroken part of Arc: any unbroken part of the circumferencethe circumference Measured in degrees or Measured in degrees or

radians as wellradians as well► Sector: a plane figure Sector: a plane figure

bounded by two radii and bounded by two radii and the included arcthe included arc

► Segment: a part cut off from Segment: a part cut off from a circle by a line, as a part a circle by a line, as a part of a circular area contained of a circular area contained by an arc and its chordby an arc and its chord


Intercepted ArcsIntercepted Arcs

►An intercepted arc is the arc that is An intercepted arc is the arc that is formed when segments intersect formed when segments intersect portions of a portions of a circle and create and create arcs.These segments in effect arcs.These segments in effect 'intercept' parts of the circle. 'intercept' parts of the circle.


Major Arcs/Minor ArcsMajor Arcs/Minor Arcs

►The bigger one is major. The smaller is The bigger one is major. The smaller is minor.minor.

►arc HN is the minor arcarc HN is the minor arc►Arc HKN is the major arcArc HKN is the major arc


TangentTangent

►a line or a plane that touches a curve a line or a plane that touches a curve at a point so that it is closer to the at a point so that it is closer to the curve in the vicinity of the point than curve in the vicinity of the point than any other line drawn through the point any other line drawn through the point


SecantSecant

►an intersecting line, esp. one an intersecting line, esp. one intersecting a curve at two or more intersecting a curve at two or more points points


ChordChord

►the line segment between two points the line segment between two points on a given curveon a given curve

►Note: a chord is inside the circle, Note: a chord is inside the circle, whereas a secant cuts through the whereas a secant cuts through the circlecircle


The Fun BeginsThe Fun Begins

►Equation:Equation:

►(a,b) are the coordinates of the center of (a,b) are the coordinates of the center of the circlethe circle

►r is the radius of the circler is the radius of the circle►For example: a circle of radius 5 centered For example: a circle of radius 5 centered

at (3,4) would have the equation:at (3,4) would have the equation:

(x-3)²+(y-4)²=25(x-3)²+(y-4)²=25


Your TurnYour Turn

►What is the equation of a circle with What is the equation of a circle with radius .6 and centered at (-27,-93)?radius .6 and centered at (-27,-93)?

►(x+27)²+(y+93)²=.36(x+27)²+(y+93)²=.36

►What are the radius and center of a What are the radius and center of a circle with the equation (x-38)²+circle with the equation (x-38)²+(y+12)²=169?(y+12)²=169?

►r=13, centered at (38,-12)r=13, centered at (38,-12)Original Content © 2008 Newport Math Club. Images adapted from mathwarehouse.com.

Intersecting Chord TheoremIntersecting Chord Theorem


Try ItTry It


Angles Formed by Angles Formed by Intersecting ChordsIntersecting Chords


►What is X?What is X?

X=1/2(arc TG+arc X=1/2(arc TG+arc ER)ER)

X=1/2(75+65)X=1/2(75+65)

X=1/2(140)X=1/2(140)

X=70X=70


►What is arc TG+arc What is arc TG+arc ER?ER?

110=1/2(arc GR+arc 110=1/2(arc GR+arc TE)TE)

220=(arc GR+arc TE)220=(arc GR+arc TE)

arc TG+arc ER=360-arc TG+arc ER=360-(arc GR+ arc TE)(arc GR+ arc TE)

arc TG+arc ER=140°arc TG+arc ER=140°Original Content © 2008 Newport Math Club. Images adapted from mathwarehouse.com.

Chord LengthChord Length

•r is the radius of the circle•c is the angle subtended at the center by the chord


►What is the length What is the length of this chord if r=3 of this chord if r=3 and c=90°and c=90°

► chord chord length=2r*sin(c/2)length=2r*sin(c/2)

► chord chord length=2*3*sin(90/2)length=2*3*sin(90/2)

► chord length=3√(2)chord length=3√(2)


Circumscribed Equilateral Circumscribed Equilateral TriangleTriangle

►What is the What is the length of each length of each side of this side of this circumscribed circumscribed equilateral equilateral triangle given triangle given that r=10?that r=10?

►What is the area What is the area of this triangle?of this triangle?


►a=b=ca=b=c►X=360/3=120X=360/3=120►a=2r*sin(x/2)a=2r*sin(x/2)►a=2*10*sin(60)a=2*10*sin(60)►a=10√(3)a=10√(3)

►Area=1/2*b*hArea=1/2*b*h►h=15h=15►Area=75√(3)Area=75√(3)

X °°


Another Way to Find Chord Another Way to Find Chord LengthLength

•r is the radius of the circle•d is the perpendicular distance from the chord to the circle center


Tangent and Secant From a Tangent and Secant From a PointPoint

►The measure of an The measure of an angle formed by a formed by a secant and a and a tangent drawn from a drawn from a point OUTSIDE the point OUTSIDE the circle is half the the is half the the difference of the intercepted arcs difference of the intercepted arcs



Two Tangents From Point Two Tangents From Point

►The measure of an angle formed by a The measure of an angle formed by a two tangents drawn from a point two tangents drawn from a point OUTSIDE the circle is half the the OUTSIDE the circle is half the the difference of the intercepted arcs difference of the intercepted arcs


Two Intersecting Secants Two Intersecting Secants From a Point From a Point

►The measure of an angle formed by a The measure of an angle formed by a two secants drawn from a point two secants drawn from a point OUTSIDE the circle is half the the OUTSIDE the circle is half the the difference of the intercepted arcs. difference of the intercepted arcs.


Conclusion?Conclusion?

►When two lines go out of a circle, the When two lines go out of a circle, the angle is angle is alwaysalways one-half the one-half the difference between the two difference between the two intercepted arcs!!!!intercepted arcs!!!!


The Intersection of a Tangent The Intersection of a Tangent and Chord and Chord

►An Angle formed by a chord and a An Angle formed by a chord and a tangent that intersect on a circle is tangent that intersect on a circle is half the measure of the intercepted half the measure of the intercepted arc arc


A Little ReviewA Little Review

►Equation: (x-a)²+(y-b)²=r²Equation: (x-a)²+(y-b)²=r²► Intersecting Chord Theorem: A*B=C*DIntersecting Chord Theorem: A*B=C*D►Chord Length: 2rsin(c/2)Chord Length: 2rsin(c/2)►Chord Length: 2√(r²-d²)Chord Length: 2√(r²-d²)►Angle Inside: Half the sum of Angle Inside: Half the sum of

intersected arcsintersected arcs►Angle Outside: Half the difference of Angle Outside: Half the difference of

intersected arcsintersected arcsOriginal Content © 2008 Newport Math Club. Images adapted from mathwarehouse.com.

Two more formulas to keep in Two more formulas to keep in mind:mind:


Tangent and Secants Tangent and Secants


Two Secants IntersectingTwo Secants Intersecting


Bonus Question: Who Bonus Question: Who introduced the symbol introduced the symbol π?π?


►William JonesWilliam Jones►A New A New

Introduction to Introduction to MathematicsMathematics

►17061706


A Great WebsiteA Great Website

►www.mathwarehouse.comwww.mathwarehouse.com►Most of the info in this powerpoint is Most of the info in this powerpoint is

from this websitefrom this website► It has a lot of reference material on It has a lot of reference material on

other topics of math as wellother topics of math as well


Documents

Newport Math Club Circles. What is a Circle? Original Content © 2008 Newport Math Club. Images adapted from mathwarehouse.com