Chapter 10 Section 3

What is a central angle? What is a major arc?

How do you find the measure of a major arc?

How do you name a major arc? What is a minor arc?

How do you find the measure of a minor arc?

How do you name a minor arc? What is a semi-circle? Two arcs are congruent

when…

central angle

minorarcmajor

arcP

B

A

C

A point H is called the midpoint of if . Any line, segment, or ray that contains H bisects .

CH

HB

CHB

CHB

What is a chord?Any segment with

endpoints that are on the circle

The endpoints of a chord are also the endpoints of an arc.

is a chord, while is formed by the

same endpoints.

ABAB

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

ABBCif and only if

AB BC

C

B

A

If a diameter (or radius) of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

E

D

G

F

DG

GF

, DE EF

From this, if one chord is

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

You can use Theorem 10.2 to find m .

AD

• Because AD DC, and . So, m = m

AD

DC

ADDC

2x = x + 40 Substitute x = 40 Subtract x from each

side.

BA

C

2x°

(x + 40)°D

Step 1: Draw any two chords that are not parallel to each other.

Step 2: Draw the perpendicular bisector of each chord. These are the diameters.

Step 3: The perpendicular bisectors intersect at the circle’s center.

center

In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.(recall distance is measured

by a perpendicular line) AB CD if and only if EF

EG.

F

G

E

B

A

C

D

AB = 8; DE = 8, and CD = 5. Find CF.

58

8 F

G

C

E

D

A

B

Because AB and DE are congruent chords, they are equidistant from the center. So CF CG. To find CG, first find DG.

CG DE, so CG bisects DE. Because DE = 8, DG = =4. 2

8

58

8 F

G

C

E

D

A

B

Then use DG to find CG.

DG = 4 and CD = 5, so ∆CGD is a 3-4-5 right triangle. 3

So CG = 3. Finally, use CG to find CF. Because CF CG, CF = CG = 3

58

8 F

G

C

E

D

A

B

Inscribed polygons-All vertices lie on

the circleContained within the

circleChords of adjacent

arcs form an inscribed polygon

Circumscribed poylgons-Contains all the

vertices of another polygon

Polygon ABCDEFGH is circumscribed about the circle

Chapter 10 Section 4

An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle.

The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle.

intercepted arc

inscribed angle

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

mADB = ½m AB

C

B

A

D

Find the measure of the blue arc.

Q

RS

T

QTS•m = 2mQRS

=•2(90°) = 180°

Find the measure of the blue arc.

ZWX•m = 2mZYX

= •2(115°) = 230° XY

Z

W

•115°

Find the measure of the blue angle.

NMP•m = ½

m•½ (100°) = 50°

NP

P

N

M

•100°

Find mACB, mADB, and mAEB.

The measure of each angle is half the measure of

m = 60°, so the measure of each angle is 30°

AB

E

D C

B

A

AB

•60

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

C D

C

B

A

D

It is given that mE = 75°. What is mF?

E and F both intercept , so E F. So, mF = mE = 75°

GH

H

G

E

F

•75°

Theater Design. When you go to the movies, you want to be close to the movie screen, but you don’t want to have to move your eyes too much to see the edges of the picture.

If E and G are the ends of the screen and you are at F, mEFG is called your viewing angle.

You decide that the middle of the sixth row has the best viewing angle. If someone else is sitting there, where else can you sit to have the same viewing angle?

Solution: Draw the circle that is determined by the endpoints of the screen and the sixth row center seat. Any other location on the circle will have the same viewing angle.

If all of the vertices of a polygon lie on a circle, the polygon is inscribed in the circle and the circle is circumscribed about the polygon.

The polygon is an inscribed polygon and the circle is a circumscribed circle.

Theorem 10.7- If an inscribed angle intercepts a semicircle, the angle is a right angle.

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.

B is a right angle if and only if AC is a diameter of the circle. A

C

B

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

D, E, F, and G lie on some circle, C, if and only if mD + mF = 180° and mE + mG = 180°

C

G

F

E

D

Find the value of each variable.

AB is a diameter. So, C is a right angle and mC = 90°

2x° = 90° x = 45

A

Q

C

B

•2x°

Find the value of each variable.

DEFG is inscribed in a circle, so opposite angles are supplementary.

mD + mF = 180° z + 80 = 180 z = 100

F

ED

G

•120°

•80°

•y°

•z°

Find the value of each variable.

DEFG is inscribed in a circle, so opposite angles are supplementary.

mE + mG = 180° y + 120 = 180 y = 60

F

ED

G

•120°

•80°

•y°

•z°