6.14.1 Arcs and Chords

The student is able to (I can):

• Apply properties of arcs

• Apply properties of chords

circle The set of all points in a plane that are a fixed distance from a point, called the center.

A circle is named by the symbol � and its center.

AAAA•

�A

diameter

radius

central angle

A line segment whose endpoints are on the circle and includes the center of the circle.

A line segment which has one endpoint on the circle and the other on the center of the circle.

An angle whose vertex is on the center of the circle, and whose sides intersect the circle.

A•

CB

D

CD is a diameterAB is a radius∠BAD is a central

angle

secant

tangent

point of tangency

A line that intersects a circle at two points

A line in the same plane as a circle that intersects it at exactly one point.

The point where the tangent and a circle intersect.

•

A

B

m

C

chord

secant

tangent

point of tangency

common tangent

A line that is tangent to two circles.

common external tangent

common internal tangent

Theorem

Theorem

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

If a line is perpendicular to a radius at a point on the circle, then it is tangent to the circle.

BELine t ⊥

Line t tangent to ⊙B •B

E

t

Theorem If two segments are tangent to a circle from the same external point, then the two segments are congruent.

•

S

A

N

D

SD ND≅

Examples The segments in each figure are tangent. Find the value of each variable.

1.

2.

•

2a + 4

5a — 32

•

6y2 18y

5a — 32 = 2a + 43a = 36a = 12

6y2 = 18yy y

6y = 18y = 3

minor arc

major arc

semicircle

An arc created by a central angle less than 180˚. It can be named with 2 or 3 letters.

An arc created by a central angle greater than 180˚. Named with 3 letters.

An arc created by a diameter. (= 180˚)

•

S

P

A

T

� �SP (or PS)is a minor arc.

� �SAP (or PAS)

is a major arc.�SPA is a semicircle.

•

•

•

Examples Find the measure of each

1. = 135˚

2. = 360 — 135

= 225˚

3. m∠CAT

•

•

135˚

F

R

ED

�mRE

�mEFR

C

A

T

260˚

= 360 — 260 = 100˚

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

ER GO⊥

•

G

E

O

R

A

GA AO≅

� �≅GR RO

Example Find the length of BU.

•

B

L

U

E

3

2

5

2 2 23 x 5+ =

x

x = 4BU = 2(4) = 8