Circles

1. Circles Chapter 9

2. Tangent Lines (9-1) • A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point. • The point where a circle and a tangent intersect is the point of tangency.

3. Tangent Lines (9-2) • Theorem: If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.

4. Tangent Lines (9-2) • Converse: If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle.

5. Tangent Lines (9-2) • Corollary: The two segments tangent to a circle from a point outside the circle are congruent. • AB = BC

6. Tangent Lines (9-1) • “Inscribed in the circle ” • “Circumscribed about the circle”

7. Tangent Lines (9-1) • Circle G is inscribed in quadrilateral CDEF. Find the perimeter of CDEF.

8. Arcs and Central Angles 9-3 • Central Angle (of a circle)- angle with its vertex at the center of the circle • Arc- unbroken part of a circle • Minor Arc (less than 180 degrees) • Name them using the endpoints • Major Arc (more than 180 degrees) • Name them using three points • Semicircles- two arcs formed by the endpoints of a diameter

9. Arcs and Central Angles 9-3 • Measure of a minor arc= measure of its central angle • Measure of a major arc= 360 degrees – measure of its minor arc • Adjacent arcs- arcs with exactly one point in common (crust of adjacent pizza slices)

10. Arc Addition Postulate • The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs • Similar to the Angle Addition Postulate

11. Congruent Arcs • Arcs in the same circle or congruent circles • Have equal measures • Arcs in two circles of different sizes cannot be congruent, even if they have the same measure (to be congruent, they must be the same shape and size)

12. Theorem 9-3 • In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent • STOP

13. Chords and Arcs (9-4) • A chord is a segment whose endpoints are on a circle. • Each chord cuts off a minor arc and a major arc

14. Chords and Arcs (9-4) • Theorem: Within a circle or congruent circles • Congruent arcs have congruent chords. • Congruent chords have congruent arcs.

15. Chords and Arcs (9-4) • Within a circle or in congruent circles…

16. Theorem 9-5 • A diameter that is perpendicular to a chord bisects the chord and its arc. Converse… • In a circle, a diameter that bisects a chord (that is not the diameter) is perpendicular to the chord. • Example

17. 86 degrees

18. Chords and Arcs (9-3) • Theorem: Within a circle or congruent circles • Chords equidistant from the center are congruent. • Congruent chords are equidistant from the center.

19. Chords and Arcs (9-4) • Find x.

20. Chords and Arcs (9-4) • Find HL and QJ. • HL= 22, QJ = 4 √3

21. Chords and Arcs (9-4) • In a circle, the perpendicular bisector of a chord contains the center of the circle. • STOP

22. Inscribed Angles (9-5) • Inscribed angle – vertex on the circle, sides of angle are chords of circle • Intercepted arc – arc formed when the sides of the inscribed angle cross the circle

23. Inscribed Angles (9-5) • Theorem: The measure of an inscribed angle is half the measure of its intercepted arc.

24. Inscribed Angles (9-5) • Find x and y. • x= ½ *(80+70) • x= 75° • m arc BC= 360- (80+70+90) = 120° • y= ½ * (70+120)= 95°

25. Inscribed Angles (9-5) • Corollary- Two inscribed angles that intercept the same arc are congruent.

26. Inscribed Angles (9-5) Corollary- An angle inscribed in a semicircle is a right angle. • GeoGebra example

27. Inscribed Angles (9-5) Corollary- The opposite angles of a quadrilateral inscribed in a circle are supplementary.

28. Inscribed Angles (9-5) • Find the value of a and b. • a= 90° • 2 *32° = 64° • b= 180- 64= 116°

29. 9-5 handout • Problems 1-9 all

30. Inscribed Angles (9-5) • The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.

31. 9-5 handout • Problems 10-21 all

32. Angle Measure and Segment Lengths (9-5) • A secant is a line that intersects a circle at two points.

33. Angle Measure and Segment Lengths (9-6) • The measure of an angle formed by two lines that intersect • inside a circle is half the sum of the measures of the intercepted arcs. • outside a circle is half the difference of the measure of the intercepted arcs.

34. The measure of an angle formed by two lines that intersectinside a circle is half the sum of the measure of the intercepted arcs • Find the measure of <1 • m<1= ½ (45 + 75) • = 60

35. The measure of an angle formed by two lines that intersectoutside a circle is half the difference of the measure of the intercepted arcs. • m <B = ½ (m AFD - m AC) • 65 = ½ (m AFD – 70) • 200 = m AFD

36. Angle Measure and Segment Lengths (9-6) • Find the value of x. • x = ½ (268 – 92) • x = 88

37. Angle Measure and Segment Lengths (9-6) • Find the value of x. • 94 = ½ (x + 122) • 188 = x + 122 • x = 66

38. Angle Measure and Segment Lengths (9-6)

39. Angle Measure and Segment Lengths (9-7)

40. Angle Measure and Segment Lengths (9-7) • Find the value of x.

41. Angle Measure and Segment Lengths (9-7) • Find the value of y.

42. Angle Measure and Segment Lengths (9-7)

43. Angle Measure and Segment Lengths (9-7)