Newport Math Club. Circles
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1 Newport Math Club Circles
2 What is a Circle?
3 Important Vocabs Center Radius (r) Diameter (d=2r) Circumference (2πr) Area (πr²) Angle Degree (360 per circle) Radian (2π per circle)
4 More Vocabs Arc Sector Tangent Secant Chord Segment
5 Arc, Sector, Segment Arc: any unbroken part of the circumference Measured in degrees or radians as well Sector: a plane figure bounded by two radii and the included arc Segment: a part cut off from a circle by a line, as a part of a circular area contained by an arc and its chord
6 Intercepted Arcs An intercepted arc is the arc that is formed when segments intersect portions of a circle and create arcs.these segments in effect 'intercept' parts of the circle.
7 Major Arcs/Minor Arcs The bigger one is major. The smaller is minor. arc HN is the minor arc Arc HKN is the major arc
8 Tangent a line or a plane that touches a curve at a point so that it is closer to the curve in the vicinity of the point than any other line drawn through the point
9 Secant an intersecting line, esp. one intersecting a curve at two or more points
10 Chord the line segment between two points on a given curve Note: a chord is inside the circle, whereas a secant cuts through the circle
11 The Fun Begins Equation: (a,b) are the coordinates of the center of the circle r is the radius of the circle For example: a circle of radius 5 centered at (3,4) would have the equation: (x-3)²+(y-4)²=25
12 Your Turn What is the equation of a circle with radius.6 and centered at (-27,-93)? (x+27)²+(y+93)²=.36 What are the radius and center of a circle with the equation (x-38)²+(y+12)²=169? r=13, centered at (38,-12)
13 Intersecting Chord Theorem
14 Try It
15 Angles Formed by Intersecting Chords
16 What is X? X=1/2(arc TG+arc ER) X=1/2(75+65) X=1/2(140) X=70
17 What is arc TG+arc ER? 110=1/2(arc GR+arc TE) 220=(arc GR+arc TE) arc TG+arc ER=360-(arc GR+ arc TE) arc TG+arc ER=140
18 Chord Length r is the radius of the circle c is the angle subtended at the center by the chord
19 What is the length of this chord if r=3 and c=90 chord length=2r*sin(c/2) chord length=2*3*sin(90/2) chord length=3 (2)
20 Circumscribed Equilateral Triangle What is the length of each side of this circumscribed equilateral triangle given that r=10? What is the area of this triangle?
21 X a=b=c X=360/3=120 a=2r*sin(x/2) a=2*10*sin(60) a=10 (3) Area=1/2*b*h h=15 Area=75 (3)
22 Another Way to Find Chord Length r is the radius of the circle d is the perpendicular distance from the chord to the circle center
23 Tangent and Secant From a Point The measure of an angle formed by a secant and a tangent drawn from a point OUTSIDE the circle is half the the difference of the intercepted arcs
24
25 Two Tangents From Point The measure of an angle formed by a two tangents drawn from a point OUTSIDE the circle is half the the difference of the intercepted arcs
26 Two Intersecting Secants From a Point The measure of an angle formed by a two secants drawn from a point OUTSIDE the circle is half the the difference of the intercepted arcs.
27 Conclusion? When two lines go out of a circle, the angle is always one-half the difference between the two intercepted arcs!!!!
28 The Intersection of a Tangent and Chord An Angle formed by a chord and a tangent that intersect on a circle is half the measure of the intercepted arc
29 A Little Review Equation: (x-a)²+(y-b)²=r² Intersecting Chord Theorem: A*B=C*D Chord Length: 2rsin(c/2) Chord Length: 2 (r²-d²) Angle Inside: Half the sum of intersected arcs Angle Outside: Half the difference of intersected arcs
30 Two more formulas to keep in mind:
31 Tangent and Secants
32 Two Secants Intersecting
33 Bonus Question: Who introduced the symbol π?
34 William Jones A New Introduction to Mathematics 1706
35 A Great Website Most of the info in this powerpoint is from this website It has a lot of reference material on other topics of math as well
36
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