Chapter 6 Circles

6.1 Circles and Related Segments and Angles

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 thelength of any radius of the circle.

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

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.

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 > rA, B, and C are points on the circle.

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.

CHORDS, DIAMETER, SECANTS

O

A

BC

D

E

FG

H

Theorem 6.1.1

A radius that is perpendicular to a chord bisects the chord.

O

A B

DC

Given: OD AB in circle O

Conclusion: OD bisects AB

Congruent Circles

Two or more circles having congruent radii are congruent circles.

o PA

B

OA PB

Concentric Circles

Concentric circles are coplanar circles that have a common center.

O

A

B

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.

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.

EXERCISES

Use the figure to identify the ff.

A

BC

DO

1. 4 chords2. 3 radii3. 1 diameter4. 1 secant line5. 2 secant rays6. An inscribed polygon7. 2 polygons not inscribed in circle O

Find the length of a circle’sdiameter for the given length ofradius.

8. 10 cm9. 3 mm10. ¾ cm11. x

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.

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.

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.

Arcs

Circles can be separated into parts called arcs ( AB, BD). A

O

D

.B

. C

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.

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.

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.

Congruent Arcs

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

Adjacent Nonoverlapping Arcs

Two arcs of a circle are adjacent nonoverlapping arcs if they have exactly one point in common.

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

mAOB = m AB

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.

Theorem 6.1.2

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.

Theorem 6.1.3

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.

Theorem 6.1.4

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.

Theorem 6.1.5

Congruent arcs have congruent chords.

O P

A

B

C

D

If AB CD, the AB CD.

Theorem 6.1.6

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

Theorem 6.1.7

Congruent chords are located the same distance from the center of the circle.

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

AACB is an inscribed angle since vertex C is a point on the circle and sides AC and CB are chords of the circle.

Theorem 6.1.8

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

BA

O

AB is a diameter of circle O.

Case 2. The diameter to the vertex of the inscribed angle lies in the interior of the angle.

A

B

C

Case 3. The diameter to the vertex of the inscribed angle lies in the exterior of the angle.

A

B

C

mABC = ½ m AC

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.

Theorem 6.1.10

If two inscribed angles intercept the same arc, then these angles are congruent.

A

B

CD

ABC intercepts AC.

ADC intercepts AC.

ABC ADC

Exercises

O E

D

C

B

A

30

45

3550

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

Exercises

RA P

C

B

DQ

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.

Exercises

Do pp 285-286 # 1-15 ½ crosswise

Homework

Do pp 286 – 287 # 16-28 even nos; 38 - 39.

One whole sheet of paper