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Circles > Formulas Assignment is Due

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Center Circle: A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. A circle with center P is called “ circle P ” or P Radius Diameter Chord Tangent Secant

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Formulas

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Standard Equation of a Circle r 2 = (x-h) 2 + (y-k) 2 Where, r = radius (h,K) = center of the circle

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Example: Write the standard equation of a circle with center (2,-1) and radius = 2 r 2 = (x-h) 2 + (y-k) 2 2 2 = (x- 2) 2 + (y- -1) 2 4 = (x-2) 2 + (y+1) 2

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Example: Give the coordinates for the center, the radius and the equation of the circle Center: Radius: Equation: Center: Radius: Equation: (-2,0) 4 4 2 =(x-(-2)) 2 +(y-0) 2 (0,2) 2 2 2 =(x-0) 2 +(y-2) 2 16=(x+2) 2 +y 2 4=x 2 +(y-2) 2

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Rewrite the equation of the circle in standard form and determine its center and radius x 2 +6x+9+y 2 +10y+25=4 (x+3) 2 (y+5) 2 + =2 2 Center: (-3,-5) Radius: 2

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Rewrite the equation of the circle in standard form and determine its center and radius x 2 -14x+49+y 2 +12y+36=81 (x-7) 2 (y+6) 2 + =9 2 Center: (7,-6) Radius: 9

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Use the given equations of a circle and a line to determine whether the line is a tangent or a secant Circle: (x-4) 2 + (y-3) 2 = 9 Line: y=-3x+6

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Example: The diagram shows the layout of the streets on Mexcaltitlan Island. 1. Name 2 secants 2. Name two chords 3. Is the diameter of the circle greater than HC? 4. If ΔLJK were drawn, one of its sides would be tangent to the circle. Which side is it?

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THM: If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. P l Q If l is tangent to circle Q at P, then

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If BC is tangent to circle A, find the radius of the circle. Use the pyth. Thm. r 2 +24 2 = (r+16) 2 r 2 +576 = (r+16)(r+16) r 2 +576 = r 2 +16r+16r+256 r 2 +576 = r 2 +32r+256 -r 2 576 = 32r + 256 -256 320 = 32r 32 10 = r A 16 24 r r BC

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Example: A green on a golf course is in the shape of a circle. A golf ball is 8 feet from the edge of the green and 28 feet from a point of tangency on the green, as shown at the right. Assume that the green is flat. 1. What is the radius of the green 2. How far is the golf ball from the cup at the center?

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Thm: If 2 segments from the same exterior point are tangent to a circle, then they are congruent. R T S P If SR and TS are tangent to circle P, then

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AB and DA are tangent to circle C. Solve for x. X 2 – 7x+20 = 8 X 2 7x+12= 0 (x-3)(x-4)=0 X=3, x=4 B D C A X 2 -7x+20 8

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Assignment

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Angle Relationships Central Inscribed Inside Outside

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Arc Length and Sector Area n= arc measure

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Find the length of Arc AB and the area of the shaded sector

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Vocabulary: 1.Minor Arc ________ 2.Major Arc _______ 3.Central Angle _______ 4.Semicircle __________ DE DBE

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Measure of Minor Arc = Measure of Central Angle Find Each Arc: a.CD _________ b.CDB ________ c.BCD _________ 148 328 180

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Measure of Minor Arc = Measure of Central Angle Find Each Arc: a.BD _________ b.BED ________ c.BE _________ 142 218 118 118

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Inscribed Angle: An angle whose vertex is on a circle and whose sides contain chords of the circle. Inscribed Angle Intercepted Arc

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Example: Find the measure of the angle Measure of Inscribed Angle = ½ the intercepted Arc 80 x x = ½ the arc x=1/2(80) x=40

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x 60 60 = ½ x x=120 Find the measure of the Arc Measure of Inscribed Angle = ½ the intercepted Arc

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Example: Find the measure of each arc or angle B A C D mADC = ______ 180 mAC = _______ 70 B A C 140

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Find the measure of

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Find m

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Example:

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Inside Angles – if two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle 1 A B D C m<1 = ½( mDC + mAB)

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Example: Find the missing angle 40 20 A B C D 1 m<1 = ½( mDC + mAB) m<1 = ½( 40+20) m<1 = ½(60) m<1 = 30

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Outside Angles 0 If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. 1 A B C m<1 = ½( mAB - mBC)

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Example: find the missing angle X = ½ (264-96) X = ½ (168) X=84 96 X 264

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