Chapter 10: Circles

10.1 Circles and Circumference

Name a circle by the letter at the center of the circle Diameter- segment that extends from one point on

the circle to another point on the circle through the center point

Radius- segment that extends from one point on the circle to the center point

Chord- segment that extends from one point on the circle to another point on the circle

Diameter=2 x radius (d=2r) Circumference: the distance around the circle

C=2πr or C= πd

chord

diameter

radius

A

B

C

D

EX

AB

DX

EC

Circle X

Diameter-

Radius-

Chord-

1. Name the circle

2. Name the radii

3. Identify a chord

4. Identify a diameter

a. Find the circumference of a circle to the nearest hundredth if its radius is 5.3 meters.

b. Find the diameter and the radius of a circle to the nearest hundredth if the circumference of the circle is 65.4 feet.

10.2 Angles, Arcs and Chords

10.2 Semi-circle: half the circle (180 degrees) Minor arc: less than 180 degrees

Name with two letters

Major arc: more than 180 degrees Name with three letters

Minor arc = central angle Arc length:

r

arclengtharc

2360

Minor arcMinor arc

Central angle

X

A

B

C

Semicircle

Minor arc = AB or BC

Semicircle = ABC or CDA

Major arc = ABD or CBD

AB + BC = 180

D

Find the value of x.

Find x and angle AZE

10.3 Arcs and Chords

If two chords are congruent, then their arcs are also congruent

In inscribed quadrilaterals, the opposite angles are supplementary

If a radius or diameter is perpendicular to a chord, it bisects the chord and its arc

If two chords are equidistant from the center of the circle, the chords are congruent

AB

C

DE

F

If FE=BC, then arc FE = arc BC

Quad. BCEF is an inscribed polygon – opposite angles are supplementary angles B and E & angles F and C

Diameter AD is perpendicular to chord EC – so chord EC and arc EC are bisected

E

A

B

C F D

X

*You can use the pythagorean theorem to find the radiuswhen a chord is perpendicular to a segment from the center

EX = FX so chords AB and CD are congruent because they are equidistant from the center

In the circle below, diameter QS is 14 inches long and chord RT is 10 inches long. Find VU.

10.4 Inscribed Angles

Inscribed angle: an angle inside the circle with sides that are chords and a vertex on the edge of the circle Inscribed angle = ½ intercepted arc

An inscribed right angle, always intercepts a semicircle

If two or more inscribed angles intercept the same arc, they are congruent

A

B

C

D

E

X

Inscribed angles:

angle BAC, angle CAD, angle DAE, angle BAD, angle BAE, angle CAE

Ex: Angle DAE = ½ arc DE

B

A

C

D

EF

G

Inscribed angle BAC intercepts a semicircle- so angle BAC =90

Inscribed angles GDF and GEF both intercept arc GF, so the angles are congruent

A. Find mX.

Refer to the figure. Find the measure of angles 1, 2, 3 and 4.

ALGEBRA Find mR.

ALGEBRA Find mI.

ALGEBRA Find mB.

ALGEBRA Find mD.

INSIGNIAS An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia shown is a quadrilateral inscribed in a circle. Find mS and mT.

10.5 Tangents

Tangent: a line that shares only one point with a circle and is perpendicular to the radius or diameter at that point.

Point of tangency: the point that a tangent shares with a circle

Two lines that are tangent to the same circle and meet at a point, are congruent from that point to the points of tangency

F

E

A

B

C

Lines AC and AF are tangent to circle X at points B and E respectively

-B and E are points of tangency

Radius XB is perpendicular to tangent AC at the point of tangency

AE and AB are congruent because they are tangent to the same circle from the same point

A. Copy the figure and draw the common tangents to determine how many there are. If no common tangent exists, choose no common tangent.

10.6 Secants, Tangents, and Angle Measures

Secant and Tangent Interior angle = ½ intercepted arc

Two Secants: Interior angle = ½ (sum of intercepted arcs)

Two Secants Exterior angle = ½ (far arc – close arc)

Two Tangents Exterior angle = ½ (far arc – close arc)

2

1

D

A

C

B

2 Secants/chords:

Angle 1 = ½ (arc AD + arc CB)

Angle 2 = ½ (arc AC + arc DB)

FB

D

E

A

C

2

1

Secant ED intersects tangent FC at a point of tangency (point F)

Angle 1 = ½ arc FE

Angle 2 = arc EA – arc FB

A. Find x.

B. Find x.

C. Find x.

A. Find mQPS.

B.

A. Find mFGI.

A.

B.

10.7 Special Segments in a Circle

Two Chords seg1 x seg2 = seg1 x seg2

Two Secants outer segment x whole secant =

outer segment x whole secant

Secant and Tangent outer segment x whole secant = tangent squared

*Add the segments to get the whole secant

AD

O

C

BG

I

F

H

E

2 chords:

AO x OB = DO x OC

2 secants:

EF x EG = EH x EI

B

D

A

C

Secant and Tangent:

AD x AB = AC x AC

A. Find x.

B. Find x.

A. Find x.

B. Find x.

Find x. Round to the nearest tenth.

Find x.

Find x.

LM is tangent to the circle. Find x. Round to the nearest tenth.

Find x. Assume that segments that appear to be tangent are tangent.