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Ch 10.1 Tangents to Circles

Ch 10.1 Tangents to Circles. Day 1 Part 1 CA Standards 7.0, 21.0. Warmup. Name 5 objects which has the shape of circle. Circle . diameter. Basic Circle. radius. Chord : a segment whose endpoints are points on the circle.

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Ch 10.1 Tangents to Circles

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  1. Ch 10.1 Tangents to Circles Day 1 Part 1 CA Standards 7.0, 21.0

  2. Warmup • Name 5 objects which has the shape of circle.

  3. Circle . diameter Basic Circle radius

  4. Chord: a segment whose endpoints are points on the circle. • Diameter: a chord that passes through the center of the circle. • Secant: a line that intersects a circle in two points. • Tangent: a line in the plane of a circle that intersects the circle in exactly one point.

  5. PS and PR are chords R Q P S

  6. Secant and Tangent k j Line j is a secant. Line k is a tangent

  7. Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter or a radius of circle. 1. AD K 2. CD B 3. EG 4. HB A D E H F G . C

  8. Tangents • Common tangent: a line or segment that is tangent to two coplanar circle

  9. Internally tangent vs. Externally tangent

  10. In a plane, the interior of a circle consists of the points that are inside the circle. • The exterior of a circle consists of the points that are outside the circle. • The point at which a tangent line intersects the circle to which it is tangent is the point of tangency.

  11. Theorems • 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 P . Q l

  12. Theorems • In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle. • If at P, then l is tangent to circle Q. P L . Q l

  13. Circles in Coordinate Geometry • Give the center and radius of each circle. Describe the intersection of the two circles and describe all common tangents.

  14. Theorem • If two segments from the same exterior point are tangent to a circle, then they are congruent. • If and are tangent to circle P, then R . P S T

  15. Using Properties of Tangents • AB is tangent to circle C at B. • AD is tangent to circle C at D. D x2 + 2 . A C 11 B

  16. C is the center of both circles, and the radii of the circles are 8 and 17. If JL is tangent to the circle of radius 8, find the length of JL.

  17. 10.2 Arcs and Chords Day 1 Part 2 CA Standards 4.0, 7.0, 16.0, 21.0

  18. Arcs Central angle A Minor arc Major arc P . B . C

  19. Measuring Arcs • The measure of a minor arc is defined to be the measure of its central angle. A mAB = 60° . 60° C B

  20. Find the measure of the arc. A = 145° mAB . 145° C B

  21. Finding Measures of Arcs • Find the measure of each arc of circle R. • MN • MPN • PMN = 80° = 280° N R P 180° . 80° 100° = 180° M

  22. Arc Addition Postulate • The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. B . C . . A mABC = mAB + mBC

  23. Find the measure of the arc. x L . N C 85° 130° M x = 145 130 + 85 + x = 360 215 + x = 360

  24. Find the measure of the arc. 2x 180 L . N C 90 x 90 x M x + x + 2x = 360 x = 90 4x = 360

  25. Theorems about Chords of Circles • In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. • If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. • If one chord is perpendicular bisector of another chord, then the first chord is a diameter.

  26. Find the measure of the arc. x+4 = 2x -x -x D 4 = x . A C 8 2x x + 4 B 8

  27. Find the measure of the arc. D . A C 2x x + 40 B

  28. Theorem • In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

  29. Pg. 599 # 9 – 48 • Pg. 607 # 12 – 47

  30. 10.3 Inscribed Angles Day 2 Part 1 CA Standards 4.0, 7.0, 16.0, 21.0

  31. Warmup • Solve 3x + 3y = 180 5x + 2y = 180

  32. An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. inscribed angle intercepted arc

  33. Measure of an Inscribed Angle • If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc. A . C D B m<ADB = ½ mAB

  34. Finding Measures of Arcs and Inscribed Angles • Find mSTQ. R S L • Q T

  35. Finding Measures of Arcs and Inscribed Angles • Find mSTQ. S W 115° Q • T

  36. Theorem • If two inscribed angles of a circle intercept the same arc, then the angles are congruent. A B C <C is congruent to <D. D

  37. Finding the Measure of an Angle • It is given that m<E = 75°. What is m<F? 75° E F

  38. Using Properties of Inscribed Polygons • If all of the vertices of a polygon lie on a circle, the polygon is inscribed in the circle and circle is circumscribed about the polygon.

  39. If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. • Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. A < B is a right angle iff AC is a diameter of the circle. • C B

  40. Find the value of the variable. Q • 2x

  41. A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. • D,E,F, and G lie on some circle iff m<D + m<F = 180° & m<E + m<G = 180° •

  42. Find the value of each variable. h° 80° • g° 120°

  43. Using an Inscribed Quadrilateral • ABCD is inscribed in circle P. • Find the measure of each angle. 2y • 3x 3y 5x

  44. 10.4 Other Angle Relationships in Circles Day 2 Part 2 CA Standards 7.0, 21.0

  45. Theorem • If a tangent and chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. B C • m<1 = ½ mAB m<2 = ½ mBCA 1 2 A

  46. Finding Angle and Arc Measures • Line m is tangent to the circle. Find m<1. m B 1 150° A

  47. Finding Angle and Arc Measures • Line m is tangent to the circle. Find the measure of the mPSR. m P R 130° • S

  48. Finding an Angle Measure • BC is tangent to the circle. Find m<CBD. C • A • B 5x 9x + 20 D

  49. Lines intersecting inside or outside a circle • If two lines intersect a circle, there are three places where the lines can intersect. • • •

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