UNIT 3Circles & Lines

SECTION 1Key Terms

WRITE DOWN EVERYTHING YOU KNOW ABOUT CIRCLES!

CHORD

Line segment that connects two points on a circle

Chords equidistant from the center are congruent

DIAMETER & RADIUS

Diameter: Chord passing through the center of the circle

Radius: Line segment from the center to the circumference (outside)

TANGENT

Line that touches a circle at exactly one point

SECANT

Line that passes through a circle at two points

What’s the difference between a chord, a tangent, and a secant?

ARC

Curve that makes up the circle Minor Arc: less than 180° Major Arc: greater than 180° Semicircle: exactly 180° (half the circle)

CENTRAL ANGLE

Angle whose vertex is on the center of the circle

Measure of a central angle is equal to the measure of the intercepted arc

EXAMPLE 2

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SECTION 2Chords

CHORDS & ARCS

Chords intercepting congruent arcs are congruent

Example: Find the measure of arc AC if arc BA = 150°, and arc BA is congruent to arc CB.

DISTANCE FROM A CHORD TO THE CENTER

Example: What is the length of BD? Hint: What shape do you see in this diagram?

SECTION 3Tangents

TANGENTS

How many tangents can you draw that touch both circles at exactly one point?

TANGENTS ARE PERPENDICULAR TO THE RADIUS THEY INTERSECT

Find the radius.

15

17

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SECTION 4Arc-Angle Relationships

INSCRIBED ANGLE Angle whose vertex is on the circle

Inscribed angle = Intercepted arc

Example: The measure of arc AC is 80°. Find the measure of AOC and ABC.

12

ANGLES INSIDE THE CIRCLE

Angle = ½ (Arc 1 + Arc 2) Arc BD = 60° and arc AC = 100 °. Find the

measure of angle AEC.

B

A

C DE

ANGLES OUTSIDE THE CIRCLE

Angle = ½ (Arc 1 – Arc 2) Chord LP is congruent to chord NM. Arc LP

measures 130°. Arc LN is three times the measure of arc PM. Find the measure of angle PQM.

M

N

L

P

Q

EXAMPLE

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SECTION 5Segment Product Theorem

ARCS BETWEEN PARALLEL LINES ARE CONGRUENT

BA

C D

Name the two congruent arcs.

SEGMENT PRODUCT THEOREM #1

LINES, not angles or arcs! Chord-Chord: AE × EB = CE × ED Example: Find x.

B

A

C DE

9

x

3 6

SEGMENT PRODUCT THEOREM #2

Tangent-Tangent: CD = AD “Hat Rule” Can you prove this?

SEGMENT PRODUCT THEOREM #3

Tangent-Secant: PA2 = PB × PC Example: If PB is 2 inches and BC is 16

inches, find PA.

SEGMENT PRODUCT THEOREM #4

Secant-Secant: BE × AE = DE × CE Example: If AB = 5, CD = 10, and DE = 12,

what is the length of BE?

D

BE

A

C

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