U NIT 3 Circles & Lines

S ECTION 1 Key Terms

W RITE DOWN EVERYTHING YOU KNOW ABOUT CIRCLES !

C HORD Line segment that connects two points on a circle Chords equidistant from the center are congruent

D IAMETER & R ADIUS Diameter: Chord passing through the center of the circle Radius: Line segment from the center to the circumference (outside)

T ANGENT Line that touches a circle at exactly one point

S ECANT Line that passes through a circle at two points What’s the difference between a chord, a tangent, and a secant?

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

C ENTRAL A NGLE Angle whose vertex is on the center of the circle Measure of a central angle is equal to the measure of the intercepted arc

E XAMPLE 2

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S ECTION 2 Chords

C HORDS & A RCS 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.

D ISTANCE FROM A C HORD TO THE C ENTER Example: What is the length of BD? Hint: What shape do you see in this diagram?

S ECTION 3 Tangents

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

T ANGENTS ARE P ERPENDICULAR TO THE R ADIUS THEY I NTERSECT Find the radius

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S ECTION 4 Arc-Angle Relationships

I NSCRIBED A NGLE 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.

A NGLES I NSIDE THE C IRCLE Angle = ½ (Arc 1 + Arc 2) Arc BD = 60° and arc AC = 100 °. Find the measure of angle AEC. B A C D E

A NGLES O UTSIDE THE C IRCLE 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

E XAMPLE

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S ECTION 5 Segment Product Theorem

A RCS B ETWEEN P ARALLEL L INES ARE C ONGRUENT B A C D Name the two congruent arcs.

S EGMENT P RODUCT T HEOREM #1 LINES, not angles or arcs! Chord-Chord: AE × EB = CE × ED Example: Find x. B A C D E 9 x 3 6

S EGMENT P RODUCT T HEOREM #2 Tangent-Tangent: CD = AD “Hat Rule” Can you prove this?

S EGMENT P RODUCT T HEOREM #3 Tangent-Secant: PA 2 = PB × PC Example: If PB is 2 inches and BC is 16 inches, find PA.

S EGMENT P RODUCT T HEOREM #4 Secant-Secant: BE × AE = DE × CE Example: If AB = 5, CD = 10, and DE = 12, what is the length of BE?

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