unit 3

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.

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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

PQ

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

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x3 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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