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Circles

Circles. > Formulas Assignment is Due. Tangent. Center. Radius. Diameter. Chord. Secant. 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. Formulas.

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Circles

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  1. Circles > Formulas Assignment is Due

  2. Tangent Center Radius Diameter Chord Secant 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

  3. Formulas

  4. Standard Equation of a Circle r2 = (x-h)2 + (y-k)2 Where, r = radius (h,K) = center of the circle

  5. Example: Write the standard equation of a circle with center (2,-1) and radius = 2 r2 = (x-h)2 + (y-k)2 22 = (x- 2)2 + (y- -1)2 4 = (x-2)2 + (y+1)2

  6. Example: Give the coordinates for the center, the radius and the equation of the circle (0,2) Center: Radius: Equation: Center: Radius: Equation: (-2,0) 2 4 22=(x-0)2+(y-2)2 42=(x-(-2))2+(y-0)2 4=x2+(y-2)2 16=(x+2)2+y2

  7. Rewrite the equation of the circle in standard form and determine its center and radius x2+6x+9+y2+10y+25=4 + =22 (x+3)2 (y+5)2 Center: (-3,-5) Radius: 2

  8. Rewrite the equation of the circle in standard form and determine its center and radius x2-14x+49+y2+12y+36=81 + =92 (x-7)2 (y+6)2 Center: (7,-6) Radius: 9

  9. 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

  10. 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?

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

  12. r A 16 r B C 24 If BC is tangent to circle A, find the radius of the circle. Use the pyth. Thm. r2+242 = (r+16)2 r2+576 = (r+16)(r+16) r2+576 = r2+16r+16r+256 r2+576 = r2+32r+256 -r2 -r2 • 576 = 32r + 256 • -256 -256 • 320 = 32r • 32 • 10 = r

  13. 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. • What is the radius of the green 2. How far is the golf ball from the cup at the center?

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

  15. X2-7x+20 B A C 8 D AB and DA are tangent to circle C. Solve for x. X2 – 7x+20 = 8 X27x+12= 0 (x-3)(x-4)=0 X=3, x=4

  16. Assignment

  17. Angle Relationships Central Inscribed Inside Outside

  18. Arc Length and Sector Area n= arc measure

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

  20. DE DBE BD Vocabulary: • Minor Arc ________ • Major Arc _______ • Central Angle _______ • Semicircle __________ <DPE

  21. Find Each Arc: • CD_________ • CDB ________ • BCD _________ Measure of Minor Arc = Measure of Central Angle 148 328 180

  22. Find Each Arc: • BD_________ • BED ________ • BE _________ Measure of Minor Arc = Measure of Central Angle 118 142 218 118

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

  24. 80 x Example: Find the measure of the angle Measure of Inscribed Angle = ½ the intercepted Arc x= ½ the arc x=1/2(80) x=40

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

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

  27. B 72 C A Find the measure of <BCA m<BCA = ______ 36

  28. B 44 A C D Find m<C M<C = 44 88

  29. Example:

  30. D m<1 = ½( mDC + mAB) C 1 A B Inside Angles – if two chords intersect in theinterior 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

  31. 20 A B 1 C m<1 = ½( mDC + mAB) 40 D Example: Find the missing angle m<1 = ½( 40+20) m<1 = ½(60) m<1 = 30

  32. A m<1 = ½( mAB - mBC) C 1 B Outside Angles • 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.

  33. 96 X Example: find the missing angle 264 X = ½ (264-96) X = ½ (168) X=84

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