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10.1 Tangents to Circles
Read pages 595 – 598 and define each term in your notes:
Circle
Center
Radius
Congruent circles
Diameter
Chord
Secant
Tangent
Tangent circles
Concentric circles
Common tangent
Interior of a circle
Exterior of a circle
Point of tangency
If a line is tangent to a circle, then it is perpendicular to the radius drawn to the
point of tangency.
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 two segments from the same exterior point are tangent to a circle, then they
are congruent.
10.2 Arcs and Chords
Read pages 603 – 606 and define the following:
Central angle
Minor arc
Major arc
Semicircle
Measure of a minor arc
Measure of a major arc
Find the measure of each arc of the circle:
AC =
BC =
ABC =
ACB =
Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the
sum of the measures of the two arcs.
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 a perpendicular bisector of another chord, then the first chord is
a diameter.
In the same circle, or in congruent circles, two chords are congruent if and only if
they are equidistant from the center.
Page 607: 1 – 11
Page 607 – 609: 12 – 52 all
607: 12 – 52
12. minor arc
13. minor arc
14. semicircle
15. minor arc
16. major arc
17. semicircle
18. major arc
19. major arc
20. 60
21. 55
22. 300
23. 305
24. 180
25. 180
26. 60
27. 65
28. 60
29. 65
30. 115
31. 120
32. 145
33. 145
34. 145
35. AC = FG = KL, ABC = FHG = KML
36. 70, 110
37. 36, 144
38. 15, 195
39. AB = BC
40. AB = CD
41. AB = AC
42. 10
43. 40
44. 170
45. 15
46. 7
47. 40
48.
49. 360 / 24 = 15
50. 90
51. 3:00 a.m.
52. The searcher is finding a chord and the
perpendicular bisector of the chord which is the
diameter. By finding the middle of the diameter
the searcher finds the center of the circle and
the location of the beacon.
10.3 Inscribed Angles
Inscribed angle: an angle whose vertex is on the circle
and whose sides are chords of the circle.
Intercepted Arc: the arc that lies on the interior of an
inscribed angle.
Measure of an Inscribed Angle
If an angle is inscribed in a circle, then its measure is half the measure of the
intercepted arc.
Find the measures of the arcs and angles.
If two inscribed angles of a circle intercept the same arc,
then the angles are congruent.
If a right triangle is inscribed in a circle, then the
hypotenuse is a diameter of the circle.
If one side of an inscribed triangle is a diameter
then the triangle is a right triangle.
A quadrilateral can be inscribed in a circle if and only if
Its opposite angles are supplementary.
Find the value of each variable.
616: 1 – 22, 24 – 29
10.4 Other Angle Relationships in Circles
If a tangent and a chord intersect at a point on a circle,
then the measure of each angle is one half the measure
of its intercepted arc.
If two chords intersect in the interior of a circle, then
the measure of each angle is one half the sum of the
measures of the arcs intercepted by the angle.
If a tangent and a secant, two tangents, or two secants
intersect in the exterior of a circle, then the measure
of the angle is one half the difference of the measures
of the intercepted arcs.
Find the indicated measure or value.
624: 8 – 34
10.5 Segment Lengths in Circles
If two chords intersect in the interior of a circle,
then part times part of one chord is equal to
part times part of the other chord.
ab = cd
If two secant segments share the same exterior endpoint,
then the outside times the whole segment of one secant is
equal to the outside times the whole segment of the other.
a(a+b)=c(c+d)
If a secant and a tangent segment share the same exterior endpoint,
then the outside times the whole secant segment is equal
to the tangent segment squared.
a ( a + b ) = c²
Examples:
10.6 Equations of Circles
The standard form of a circle with a radius r and center (h,k) is
(x-h)² + (y-k)² = r²
Write the equation of the circles with the following:
Center (4,3), r = 5
center (-2,2), r = 3
center (-5,-6), r = 1
center (0,0), r = 8
Give the center and radius of the circle, and then sketch the graph:
(x-7)²+ (y-4)² = 36
(x+6)²+ (y-3)² = 81
x²+ (y+5)² =1
Give the coordinates of the center, the radius, and the equation of the circle.
Class work: 638: 7 – 29, 33 - 40 all
x²+ y² = 12
MAD Quiz 10.6
Write the equation of the circles with the following
information.
1. center ( 2,3), r = 3
2. center (3,-4), r = 5
3. center (-2,-7), r = 2
4. center (0,5), r = 9
5. Sketch the graph of the circle: (x-2)² + (y+3) ² = 4
Central angle = arc
Interior angle = ½(arc + arc)
Inscribed angle = ½ arc
Exterior angle = ½(arc – arc)
Chords: part x part = part x part
Secants: outside x whole = outside x whole
Secant and tangent: outside x whole = outside²