Powerpoint presentation about circle

1. CIRCLES
CIRCLES AND RELATED SEGMENTS AND ANGLES

2. DEFINITIONS
• A circle is the set of all points in a plane that
are a given distance from a given point
called the center.
• The given distance, r, is the
length of any radius of the circle.

3. • A radius is a segment extending from the center to
any point on the circle.

4. INTERIOR/EXTERIOR
• The interior of circle O is the set of all points I in the
plane of the circle such that OI < r.
• The exterior of circle O is the set of all points E in the
plane of the circle such that OE > r.

5. 
O A
B
C
D
E
F
G







OA, OB, OC radii of
circle O
OA  OB  OC
D and G are points in
the interior of the
circle.
OD < r; OG < r
E and F are points in the
exterior of the circle.
OE > r; OF > r
A, B, and C are points on
the circle.

6. • A chord is a segment that joins two points on the
circle.
• A diameter is any chord that contains the center.
• A secant is any line, ray, or segment that contains a
chord.

7. CHORDS, DIAMETER, SECANTS
 O
A
B
C
D
E
F
G
H

8. THEOREM 6.1.1
• A radius that is perpendicular to a chord bisects the
chord.
 O
A B
D
C
Given: OD  AB in circle O
Conclusion: OD bisects AB

9. CONGRUENT CIRCLES
• Two or more circles having congruent radii are
congruent circles.
 o  P
A
B
OA  PB

10. CONCENTRIC CIRCLES
• Concentric circles are coplanar circles that have a
common center.
 O
A
B

11. SPHERE
• A sphere is the set of all points in space that are a
given distance from a given point.
• Every sphere has a center, interior and exterior points,
radii, diameters, chords, and secants.

12. • If a plane intersects a sphere in more than
one point, then the intersection is a circle.
• If the sphere’s center is a point of the
plane, then the intersection is a great circle.

13. EXERCISES
Use the figure to identify the ff.
A
B
C
D
O


14. 1. 4 chords
2. 3 radii
3. 1 diameter
4. 1 secant line
5. 2 secant rays
6. An inscribed polygon
7. 2 polygons not inscribed in circle O

15. Find the length of a circle’s
diameter for the given length of
radius.
8. 10 cm
9. 3 mm
10. ¾ cm
11. x

16. True of false?
16. If a segment is a chord of a circle, then it is also a
diameter.
17. If a segment is a diameter of a sphere, then it is
also a chord.
18. If a segment is a radius of a circle, then it is also a
chord.

17. 19. If two circles are concentric, then their radii are
congruent.
20. If two circles are congruent, then their diameters
are congruent.
21. A sphere has exactly two diameters.

18. 22. If two spheres have the same center, then they are
congruent.
23. If AB is a chord of a sphere, then AB is also a secant
of the sphere.
24. If AB is a secant of a circle, then AB is also a chord
of the circle.

19. ARCS
•Circles can be separated into
parts called arcs
( AB, BD).
A
O
D
.
B
. C


20. SEMICIRCLE
• When the endpoints of an arc are also the endpoints
of a diameter, the arc is a semicircle.
• The measure of a semicircle is 180.
• When an arc is not a semicircle, it is either a minor arc
or a major arc.

21. NAMING ARCS
• A minor arc is named using two letters that
correspond to the endpoints of the arc.
• A major arc is named using three letters.
• A semicircle is named using three letters.

22. CENTRAL ANGLE
• An angle is a central angle of a circle if its vertex is the
center of the circle.
• The measure of a minor arc is the measure of the
central angle.
• The measure of a major arc is the difference between
the measure of its related minor arc and 360.

23. CONGRUENT ARCS
• In the same circle or in congruent circles, two arcs are
congruent if and only if they have equal measures.

24. ADJACENT
NONOVERLAPPING ARCS
• Two arcs of a circle are adjacent nonoverlapping arcs if
they have exactly one point in common.

25. POSTULATE 16
• Central Angle Postulate
In a circle, the degree measure of a central angle is
equal to the degree measure of its intercepted arc.
O 
A
B
mAOB = m AB

26. POSTULATE 17 ARC ADDITION
POSTULATE
• The measure of an arc formed by two adjacent
nonoverlapping arcs is the sum of the measures of
those two arcs.
• If AB and BC intersect only at point B, then mAB +
mBC = mABC.

27. THEOREM
• Congruent minor arcs of congruent circles or the same
circle have congruent central angles.
O  P 
A
B
C
D
If AB  CD, then
AOB  CPD.

28. THEOREM
• In a circle (or in congruent circles), congruent central
angles have congruent arcs.
 O  P
A
B
C
D
If AOB  CPD.
then AB  CD.

29. THEOREM
• In a circle (or in congruent circles), congruent chords
have congruent minor (major) arcs.
 O  P
A
B
C
D
If AB  CD, the AB  CD.

30. THEOREM 6.1.5
• Congruent arcs have congruent chords.
 O  P
A
B
C
D
If AB  CD, the AB  CD.

31. THEOREM
• Chords that are at the same distance from the center
of a circle are congruent.

A
B
C
D
O
E
F
Given: Chords CD and EF
are of the same distance
from O.
Conclusion: CD  EF

32. THEOREM 6.1.7
• Congruent chords are located the same distance from
the center of the circle.

33. INSCRIBED ANGLE
• An angle is called an inscribed angle of a circle if and
only if its vertex is on the circle and its sides contain
chords of the circle.
C
B
A
ACB is an inscribed angle
since vertex C is a point on
the circle and sides AC and
CB are chords of the circle.

34. THEOREM
• The measure of an inscribed angle of a circle is one-half the
measure of its intercepted arc.
• Case 1 One side of the inscribed angle is a diameter.
C
B
A

O
AB is a diameter of circle O.

35. • Case 2. The diameter to the vertex of the inscribed
angle lies in the interior of the angle.
A
B
C

36. • Case 3. The diameter to the vertex of the inscribed
angle lies in the exterior of the angle.
A
B
C
mABC = ½ m AC

37. THEOREM 6.1.9
• An inscribed angle in a semicircle is a right angle.
A
B
C
AC is a diameter.
ABC is a semicircle.
ABC is a right angle.

38. THEOREM
• If two inscribed angles intercept the same arc, then
these angles are congruent.
A
B
C
D
ABC intercepts AC.
ADC intercepts AC.
ABC  ADC

39. EXERCISES

O E
D
C
B
A
30
45
35
50
1. Explain why there are no
congruent chords.
2. List all the minor arcs and
their measures.
3. Starting with AE and
ending with AB, list 4
chords with endpoint A in
order of their distance
from the center O.
4. List all the all chords in
order from longest to
shortest.
Note: Discuss
at the end of
the lesson

40. EXERCISES
 R
A P
C
B
D
Q
1. If RP  RQ, then AB__CD and
AB__CD.
2. If RP  RQ, then CQ__AP and
AB__CD.
3. If RP > RQ, then AB__CD.
4. If RP > RQ, the CQ__AP.
5. If CD < AB, then RQ__RP
6. If CD < AB, then CQ__AP.
7. If CQ = 5 and RQ = 12, find the
length of any radius.

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