Circles

1
Circles

• Chapter 6

2
Essential Questions

• How do I identify segments and lines related to
  circles?
• How do I use properties of a tangent to a circle?

3
Definitions

• A circle is the set of all points in a plane that
  are equidistant from a given point called the
  center of the circle.
• Radius the distance from the center to a point
  on the circle
• Congruent circles circles that have the same
  radius.
• Diameter the distance across the circle through
  its center

4
Diagram of Important Terms
center
5
Definition

• Chord a segment whose endpoints are points on
  the circle.

6
Definition

• Secant a line that intersects a circle in two
  points.

7
Definition

• Tangent a line in the plane of a circle that
  intersects the circle in exactly one point.

8
Example 1

• Tell whether the line or segment is best
  described as a chord, a secant, a tangent, a
  diameter, or a radius.

tangent
diameter
chord
radius
9
Definition

• Tangent circles coplanar circles that intersect
  in one point

10
Definition

• Concentric circles coplanar circles that have
  the same center.

11
Definitions

• Common tangent a line or segment that is
  tangent to two coplanar circles
• Common internal tangent intersects the segment
  that joins the centers of the two circles
• Common external tangent does not intersect the
  segment that joins the centers of the two circles

12
Example 2

• Tell whether the common tangents are internal or
  external.

a.
b.
common internal tangents
common external tangents
13
More definitions

• Interior of a circle consists of the points
  that are inside the circle
• Exterior of a circle consists of the points
  that are outside the circle

14
Definition

• Point of tangency the point at which a tangent
  line intersects the circle to which it is tangent

15
Perpendicular Tangent Theorem

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

16
Perpendicular Tangent Converse

• 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.

17
Definition

• Central angle an angle whose vertex is the
  center of a circle.

18
Definitions

• Minor arc Part of a circle that measures less
  than 180
• Major arc Part of a circle that measures
  between 180 and 360.
• Semicircle An arc whose endpoints are the
  endpoints of a diameter of the circle.
• Note major arcs and semicircles are named with
  three points and minor arcs are named with two
  points

19
Diagram of Arcs
20
Definitions

• Measure of a minor arc the measure of its
  central angle
• Measure of a major arc the difference between
  360 and the measure of its associated minor arc.

21
Arc Addition Postulate

• The measure of an arc formed by two adjacent arcs
  is the sum of the measures of the two arcs.

22
Definition

• Congruent arcs two arcs of the same circle or
  of congruent circles that have the same measure

23
Arcs and Chords Theorem

• In the same circle, or in congruent circles, two
  minor arcs are congruent if and only if their
  corresponding chords are congruent.

24
Perpendicular Diameter Theorem

• If a diameter of a circle is perpendicular to a
  chord, then the diameter bisects the chord and
  its arc.

25
Perpendicular Diameter Converse

• If one chord is a perpendicular bisector of
  another chord, then the first chord is a
  diameter.

26
Congruent Chords Theorem

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

27
Right Triangles Pythagorean Theorem
Radius is perpendicular to the tangent. ? lt E is
a right angle
 
 
28
Example 3
Use the converse of the Pythagorean Theorem to
see if the triangle is right.
112 432 ? 452
121 1849 ? 2025
1970 ? 2025
29
Congruent Tangent Segments Theorem

• If two segments from the same exterior point are
  tangent to a circle, then they are congruent.

30
Example 4
31
Example 1

• Find the measure of each arc.

70
360 - 70 290
180
32
Example 2

• Find the measures of the red arcs. Are the arcs
  congruent?

33
Example 3

• Find the measures of the red arcs. Are the arcs
  congruent?

34
Example 4
35
Definitions

• Inscribed angle an angle whose vertex is on a
  circle and whose sides contain chords of the
  circle
• Intercepted arc the arc that lies in the
  interior of an inscribed angle and has endpoints
  on the angle

36
Measure of an Inscribed Angle Theorem

• If an angle is inscribed in a circle, then its
  measure is half the measure of its intercepted
  arc.

37
Example 1

• Find the measure of the blue arc or angle.

a.
b.
38
Congruent Inscribed Angles Theorem

• If two inscribed angles of a circle intercept the
  same arc, then the angles are congruent.

39
Example 2
40
Definitions

• Inscribed polygon a polygon whose vertices all
  lie on a circle.
• Circumscribed circle A circle with an inscribed
  polygon.

The polygon is an inscribed polygon and the
circle is a circumscribed circle.
41
Inscribed Right Triangle Theorem

• 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.

42
Inscribed Quadrilateral Theorem

• A quadrilateral can be inscribed in a circle if
  and only if its opposite angles are supplementary.

43
Example 3

• Find the value of each variable.

b.
a.
44
Tangent-Chord Theorem

• If a tangent and a chord intersect at a point on
  a circle, then the measure of each angle formed
  is one half the measure of its intercepted arc.

45
Example 1
46
Try This!
47
Example 2
48
Interior Intersection Theorem

• 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 and its vertical angle.

49
Exterior Intersection Theorem

• If a tangent and a secant, two tangents, or two
  secants intersect in the exterior of a circle,
  then the measure of the angle formed is one half
  the difference of the measures of the intercepted
  arcs.

50
Diagrams for Exterior Intersection Theorem
51
Example 3

• Find the value of x.

52
Try This!

• Find the value of x.

53
Example 4

• Find the value of x.

54
Example 5

• Find the value of x.

55
Chord Product Theorem

• If two chords intersect in the interior of a
  circle, then the product of the lengths of the
  segments of one chord is equal to the product of
  the lengths of the segments of the other chord.

56
Example 1

• Find the value of x.

57
Try This!

• Find the value of x.

58
Secant-Secant Theorem

• If two secant segments share the same endpoint
  outside a circle, then the product of the length
  of one secant segment and the length of its
  external segment equals the product of the length
  of the other secant segment and the length of its
  external segment.

59
Secant-Tangent Theorem

• If a secant segment and a tangent segment share
  an endpoint outside a circle, then the product of
  the length of the secant segment and the length
  of its external segment equals the square of the
  length of the tangent segment.

60
Example 2

• Find the value of x.

61
Try This!

• Find the value of x.

62
Example 3

• Find the value of x.

63
Try This!

• Find the value of x.