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Review Sheet Chapter Ten

100°. 120°. w. 9. y. 5. 3. 10. 3. 2. z. x. 10. 60°. 50°. x. 12. 20°. z. 12. Review Sheet Chapter Ten. Equation of a Circle : ( x – h ) 2 + ( y – k ) 2 = r 2 where ( h , k ) is the center and r is the radius Circle Parts:

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Review Sheet Chapter Ten

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  1. 100° 120° w 9 y 5 3 10 3 2 z x 10 60° 50° x 12 20° z 12 Review Sheet Chapter Ten Equation of a Circle: (x – h) 2 + (y – k)2 = r2 where (h, k) is the center and r is the radius Circle Parts: Chord is a segment going from edge to edge; Diameter is chord that goes through the center (longest chord) Radius goes from center to the edge of the circle (its one-half the diameter) Radii are perpendicular to tangents (Pythagorean Theorem) Tangents from a single point (outside the circle) are equidistant Arc length is a percentage (central angle / 360) of the circumference Circle Angles: Circle Segments: Test Taking Tips: Check and see if the angles and distances make sense compared to the given in the picture r r y Exterior  z = 50°- 20° = 30° = 15° 2 2 Interior w = 100°+120° = 220° = 110° 2 2 Central  x = 60° Inscribed  y = ½ (60) = 30° “outside part times whole thing ” 3(3 + y) = 2*(2 + 10) 9 + 3y= 24 3y = 15 y = 5 2 “outside part times whole thing = tangent squared” 3(3 + 9) = z2 36 = z2 6 = z to get “x” 5*x = 2*10 5x = 20 x = 4

  2. Polygons and Circles • SSM: • arc BC + 140 must be half the circle Arc AB = 140, then so does DC = 140if arc BC = x, then arc AD = x as well Once around the circle is 360 so 360 = 280 + 2x 80 = 2x 40 = x = arc BC

  3. Equation of a circle: (x – h)2 + (y – k)2 = r2 where (h, k) is the center and r is the radius. First use distance formula to find length of diameter: d = (-7 – 1)2 + (-4 – 2)2 = 64 + 36 = 10 So r = 5 and r2 = 25 Then use midpoint formula to find center of circle: MP =( (-7 + 1)/2 , (-4 + 2)/2 ) = (-3, -1) So center (h, k) is at (-3, -1)  (x – (-3))2 + (y – (-1))2 = 25 Drop and drag pieces into place.: (x + 3)2 + (y + 1)2 = 25

  4. Ch 10 Coordinate Relations and Transformations • SSM: • find equation on equation sheet Use equation from the equation sheet and substitute the center into equation. Remember that the negative of a negative is a positive.

  5. Ch 10 Coordinate Relations and Transformations • SSM: • y is measure of a small medium acute angle • Eliminate C and D. Three angles in a triangle add to 180. Vertical angles give us Angle TRS = 180 – (50+92) = 38. Angle SQT = y and shares same arc as angle TRS so they must have the same measures. y = 38.

  6. Ch 10 Coordinate Relations and Transformations • SSM: • find formula on formula sheet • diameter = twice radius • not much help Point on a circle much satisfy the equation of a circle. Put given information (remember radius = ½ diameter) into equation. (x –(-2))2 + (y – (-2))2 = 52 (x + 2)2 + (y + 2)2 = 25 Plug points into equation and see which satisfies

  7. Ch 10 Coordinate Relations and Transformations • SSM: • slightly less than ¼ Circumference Arc length is found by the following proportion: 70 x --------- = --------- 360x = 140(40) 360x = 5600 360 C=2r x = 48.87

  8. Ch 10 Coordinate Relations and Transformations • SSM: • find formula • decipher equation Equation of a circle: (x – h)2 + (y – k)2 = r2 (x + 4)2 + (y – 5)2 = 32 Remember negative of a negative is a positive. Center (-4, 5)

  9. Ch 10 Coordinate Relations and Transformations • SSM: • if figure is to scale • 2 < x < 5 by eyes Segments inside the circle parts multiplied are equal: 2(3x – 2) = 5(x) 6x – 4 = 5x x – 4 = 0 x = 4 Plug back in to get KZ = 3x – 2 + 2 = 3x = 3(4) = 12

  10. Ch 10 Coordinate Relations and Transformations • SSM: • plug in given values • find r Plug in the values of the point for x and y into circle equation: (x – 1)2 + (y – 3)2 = r2 (4 – 1)2 + (7 – 3)2 = r2 (3)2 + (4)2 = r2 9 + 16 = r2 25 = r2 5 = r

  11. Ch 10 Coordinate Relations and Transformations (x – 4)² • SSM: • find formula on formula sheet (y + 7)² + 4² Circle equation from formula sheet: (x – h)2 + (y – k)2 = r2 center (h, k) Remember a negative of a negative is a positive.

  12. Ch 10 Coordinate Relations and Transformations • SSM: • eyes tell us arc is longer than the minute hand Arc length is given by the following proportion: 108 x ------ = ----------- 216(10) = 360x 6785.84 = 360x 18.84 = x 360 C = 2r

  13. Ch 10 Coordinate Relations and Transformations • SSM: • plug in answers Point on the circle must satisfy the equation. (x – 1)2 + (y – 3) 2 = 72 (8 – 1) 2 + (3 – 3) 2 = 72 72 + 02 = 72

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