UNIT 3 Circles & Lines
Dec 27, 2015
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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