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Circles

Circles. 10-1 Circles. C. I. Definitions Circle The set of all points in a plane that are at a given distance from a given point in that plane .  Symbol ○R Radius The distance between the center of a circle and any point on the edge of the circle. AR,BR

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Circles

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  1. Circles

  2. 10-1 Circles C • I. Definitions • Circle The set of all points in a plane that are at a given distance from a given point in that plane.  • Symbol ○R • Radius The distance between the center of a circle and any point on the edge of the circle.AR,BR • Diameter The distance across the circle that goes through the radiusAB • Chord A segment that goes from one side of a circle to another. AC • Circumference the distance around the circleC= 2  r • Total degrees 360 B A R

  3. II. EXAMPLES • 1. Find the circumference if • a. r = 8 b. d = 12

  4. 2. Find d and r if C = 136.9 cm.

  5. 3. Circle A radius=8, Circle B radius = 14, and JE =4. Find EB and DC. D C J A E B

  6. I. Central Angle an angle whose vertex contains the center of a circle. 10-2 Angles and Arcs

  7. II. Arc Part of a circle; the curve between two points on a circle.

  8. If circle is divided into two unequal parts or arcs, the shorter arc (in red) is called the minor arc and the longer arc (in blue) is called the major arc. A Minor arc- 2 letters Major arc- 3 letters B C

  9. III. Semicircle • a semicircle is an arc that makes up half of a circle 180°

  10. Arc Addition Postulate - The measures of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.   That is, if B is a point on     , then    +    =    .

  11. IV. ARC MEASURE The measure of a minor arc = central angle.  The measure of a major arc = 360 minus the measure of its central angle.

  12. V. ARC LENGTH LENGTH OF THE ARC is a part of the circumference proportional to the measure of the central angle when compared to the entire circle

  13. VII. CONCENTRIC CIRCLES CONCENTRIC CIRCLES lie in the same plane and have the same center, but have different radii.  ALL CONCENTRIC CIRCLES ARE SIMILAR BC ALL CIRCLES ARE SIMILAR!

  14. VIII. CONGRUENT ARCS TWO ARCS WITH THE SAME MEASURE AND LENGTH

  15. Example 1: Find the length of arc RT and the measurement in degrees.

  16. a. Find the length of arcs RT and RSTb. Find the measurement in degrees of both.

  17. 3. Find the arc length of RT and the degrees measurement of RT.

  18. If <NGE  < EGT, <AGJ =2x, <JGT = x + 12, and AT and JN are diameters, find the following:a. x b. m NE c. m JNE A J E N T

  19. 5.

  20. 6. Find x. N M 9x 8x Q A 19x O R

  21. I. Arc of the chord When a minor arc and a chord share the same endpoints, we called the arc the ARC OF THE CHORD. 10-3 Arcs and Chords

  22. II. Relationships • 2 minor arcs are  if their chords are .

  23. If a diameter is perpendicular to a chord, it bisects the chord and the arc.

  24. 2 chords are  if they are equidistant from the center

  25. Inscribed polygons must have vertices on the circle.

  26. 1. Circle N has a radius of 36.5 cm. Radius is perpendicular to chord FG, which is 53 cm long. • a. If m FG= 85, find m HG. • b. Find NZ.

  27. 2. Chords FG and LY are equidistant from the center. If the radius of M is 32, find FG and BY. FG = 46.4 LY = 23.2

  28. 3. mWX = 30, mXY = 50, mYZ = 30. WY= 14, FIND XZ.

  29. 4. RT is a diameter. If US = 9, find SV.

  30. XZ= 12, UV = 8, WY is a diameter.Find the length of a radius.

  31. 6. IF AB and DC are both parallel and congruent and MP = 7, find PQ.

  32. 10-4 Inscribed Angles • I. Definitions • Inscribed angle — An angle that has its vertex on the circle and its sides are chords of the circle • Intercepted arc — An intercepted arc is the arc that lies "inside" of an inscribed angle

  33. If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc

  34. If two inscribed angles of a circle or congruent circles intercept congruent arcs or the same arc, then the angles are congruent

  35. If an inscribed angle of a circle intercepts a semicircle, then the angle is a right angle

  36. If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary

  37. 1.

  38. 2.

  39. 3. If mLM=120, mMN=45, and mNQ=105, find the numbered angles. < 1= 22.5 < 2 = 60 < 3 = 45 < 4 = 22.5 < 5 = 112.5

  40. 4. If <2= 3a + 2 and < 3= 12 a – 2, find the measures of the numbered angles m1 = 45, m4 = 45 <2 = 20, < 3= 70

  41. 5. If mW = 74 and mZ = 112, find mY and mX. • = mX • 106 = mY

  42. 10-5 Tangents • I. A line is TANGENT to a circle if it intersects the circle in EXACTLY ONE point.  This point is called the POINT OF TANGENCY.

  43. If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

  44. II. Common External Tangents & Common Internal Tangents A line or line segment that is tangent to two circles in the same plane is called a common tangent COMMON EXTERIOR

  45. COMMON INTERIOR

  46. If two segments from the same exterior point are tangentto the circle, then they are congruent

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