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Chapter 8 Analyzing Circles

Chapter 8 Analyzing Circles. Find the degree and linear measure of an arc Find measures of angles in circles Use properties of chords, tangents, and secants Write equations of circles. Exploring Circles. Definitions Circle – set of all points in a plane equidistant from the center point.

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Chapter 8 Analyzing Circles

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  1. Chapter 8Analyzing Circles Find the degree and linear measure of an arc Find measures of angles in circles Use properties of chords, tangents, and secants Write equations of circles

  2. Exploring Circles Definitions • Circle – set of all points in a plane equidistant from the center point. • Chord – segment with endpoints on the circle. • Diameter – chord that goes through the center point. • Radius – segment that connects the center to a point on the circle. • Secant – line that intersects the circle at two points. • Tangent – line that intersects the circle at one point. • Circumference of a circle – • Area of a circle – A=r2 • Revolution – equals circumference

  3. C is the center point. • The circle is called Circle C • Chord – AB • Diameter – JK • Radius – CE • Secant - EG • Tangent – JH • J is called the point of tangency

  4. Examples Identify the segment • AB • BD • BC • CD • ST • Point of tangency radius diameter B secant S A chord tangent Q C Q T D

  5. Word problem example • The wheel of a lawn mower made 32 revolutions while cutting a strip of grass. If the diameter of a wheel is 8 inches, how far did the lawn mower travel? Circumference = 2r = 2 x 3.14 x 4 = 25.12 Multiply 25.12 x 32 = 803.84 inches

  6. Complete the table

  7. Practice • The circumference of Earth is 25,000 miles. What is the Earth’s approximate diameter? • If the circumference of a circle is 24 meters, find the radius. • If d = 5, find C. • If a 24-inch bicycle makes 1000 revolutions, how far will the bicycle travel? 7957.75 miles 3.8 meters 52 About 75,398 inches or 1.2 miles

  8. Angles and Arcs • A circle = 360° • Minor arc < 180° • Major arc > 180° • Semicircle = 180° • A central angle has the center point as its vertex. • A central angle is equal to its arc. • The length of an arc is central angle x  x d 360 2r

  9. Examples • AC is a __________ • ADC is a _________ • mAC = ______ • mADC = _______ Minor arc Major arc 60 degrees D 300 degrees . B 60 A C

  10. Examples continued 1. mACD = ______ 2. mAD = _____ 3. Length of BC = ____ 4. Length of ACD = ____ 215° cm 145° 5.7cm 18.8cm

  11. Concentric circles – lie in the same plane and have the same center. • Congruent circles – have the same radius • Similar circles – all circles are similar.

  12. Warm Up • m∠APE = • m ∠ CPB = • mEA = • mCE = • mCAB = • mEDB = • If CB = 8in, find lengths CE, AC, CAB 50° 180° 50° 40° 180° 140° 2.8in 6.3in 12.6in

  13. Arcs and Chords • Two arcs are congruent if their chords are congruent. • If a diameter is perpendicular to a chord, then it bisects the chord and its arc. • Two chords are congruent if they are equidistant from the center point. • A polygon is inscribed if each of its vertices lies on a circle.

  14. Warm Up • Name a segment congruent to JM. • Name an arc congruent to (arc)KL. • Name an arc congruent to (arc)JN. • Name a segment congruent to JB. • Name the midpoint of JL. LM JK LN PB M

  15. Inscribed Angles • The measure of an inscribed angle is ½ the measure of its arc. • If two inscribed angles intercept the same arc, then they are congruent. • If the inscribed angle intercepts a semicircle, then it is a right angle. • If a quadrilateral is inscribed in a circle then opposite angles are supplementary.

  16. Examples 1. ∠BCD = 2. arc(FH) = 140 degrees 30 degrees C G 70 F H B D 60

  17. More examples 80 1. x = __ y = __ 2. g = ___ h = ___ 90 30 105 A h g 60 C x 100 75 y B

  18. More Examples X + 12 Find x 1. 2. 3. 4. X = 6 C D G X = 15 52 A 3x 2x - 4 B F E 4x - 5 K N X = 23.75 H 3x - 4 J L I X = 35 M 2x + 9

  19. Tangents • A tangent line is perpendicular to the radius at the point of tangency. • 2 segments that meet at the same point outside the circle are congruent. E CB perp. DB C G D F B EG = FG

  20. Continued • Common external tangents • Common internal tangents D C F E

  21. Examples 8 28 • DE = ___ 2. radius = ____ D H 17 I J F E 15 K HK = 53 KI = 25

  22. One more example H X = 7 3x + 2 23

  23. Secants, tangents and angle measures • If two secants intersect inside the circle, then the measure of the angle formed is ½ (sum of the measures of the arcs) • If two secants intersect outside the circle then the measure of the angle formed is ½ (difference of the measures of the arcs) • If a secant and a tangent intersect at the point of tangency, then the angle is ½ (measure of its arc)

  24. Diagrams ½(74 + 24) = 49 24 49 74

  25. Diagrams continued • ½ (96 – 36) = 30 30 96 36

  26. Diagrams continued • ½ (280) = 140 280 140

  27. Chapter 7 Study Guide Define or draw a diagram • Circle • Radius • Diameter • Chord • Secant • Tangent • Find the circumference of a circle

  28. Study Guide continued • Find measures of arcs given central angles • Find measures of central angles given arcs • Find measures of inscribed angles given arcs • Find measures of arcs given inscribed angles • Find values of x using theorems in 9.5 and 9.6

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