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**By Mark Hatem and Maddie Hines**

Now what you’ve all been waiting for.. Chapter 12:Circles By Mark Hatem and Maddie Hines

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The radius of a circle is a line segment that connects the center of the circle to any point on the same circle. All radii are equal A chord is a line segment that connects two points of the circle. A diameter is a chord that contains the center of the circle. Lesson 1

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**If a line through the center of a circle is perpendicular to a chord it also bisects the chord.**

If a line through the center of a circle bisects a Chord that is not the diameter, it is perpendicular to the chord Chords in Lesson 1

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**The perpendicular bisector of a chord contains the center of a circle.**

One more Chord….

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A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point. Theorem 59: If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of contact. Theorem 60: If a line is perpendicular to a radius at its outer endpoint, it is tangent to the circle. Section 2

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**Lesson 3 – Angles and Arcs**

A central angle is an angle whose vertex is the center of the circle. A reflex angle is an angle whose measure is more than 180°. The degree measure of an arc is the measure of its central angle. The arc postulate: If C is on arc AB then mAC + mCB = mAB Lesson 3 – Angles and Arcs

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**Chords in Lesson 3 In a circle equal chords have equal arcs.**

In a circle equal arcs have equal chords.

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An inscribed angle is an angle whose vertex is on a circle, with each of the angle’s sides intersecting the circle in another point Theorem 63: An inscribed angle is equal in measure to half its intercepted arc. Corollary 1 to Theorem 63: Inscribed angles that intercept the same arc are equal. Corollary 2 to Theorem 63: An angle inscribed in a semicircle is a right angle. Section 4

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Lesson 5 - Secants A secant is a line that intersects a circle in two points. A secant angle is an angle whose sides are contained in two secants of a circle so that each side intersects the circle in at least one point other than the angle’s vertex.

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**Calculating Secant Angles**

A secant angle whose vertex is inside a circle is equal in measure to half the sum of the arcs intercepted by it and its vertical angle. A secant angle whose vertex is outside a circle is equal in measure to half the difference of its larger and smaller intercepted arc. 150° 89° (120+89)/2=104.5 (150-10)/2=70 120° 10°

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If a line is tangent to a circle, then any segment of the line having the point of tangency as one of its end points is a tangent segment to the circle. Theorem 66, The Tangent Segments Theorem: The tangent segments to a circle from an external point are equal. Theorem 67, The Intersecting Chords Theorem: If two chords intersect in a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. Section 6

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Intersecting Secants Theorem: If two secants intersect outside of a circle, for each secant, the product of the segment outside the circle and the entire secant are equal. A(a + b) = c(c + d) The Intersecting Secant-Tangent Theorem: If a tangent and a secant intersect outside of a circle, then the square of the lengths of the tangent is equal to, for the secant, the product of the segment outside the circle and the entire secant. T2= a(a + b) Additional lessons

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**Angles and Arcs in Circles Notes**

Central Angle x°=a° Angle inscribed in a semicircle x°=90° Inscribed angle x°=1/2a° Angles inscribed in the same arc x°=y°=1/2a° Tangent and radius angle Tangent and chord angle a x x a a x x y a x x

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**Angles and Arcs in Circles Notes Cnt.**

Two Chords Angle x°=a°+b° 2 Two tangents x°=(b-a)/2 Or x°=180-a Or x°=180-b Two Secants x°= b-a Secant and Tangent a b x x a b b b x a a X

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#2 Section 2: Know the definition of tangent and know that it creates a right angle with the radius as this will be helpful in problem solving #1 Section 4: Inscribed angles are one of the most important concepts in the chapter and is found in many problem solving questions and the corollaries are helpful shortcuts #3 Section 6: Two helpful simple formulas were given #4 Additional Lesson: Two more formulas are given with regards to secants and tangents What's Important

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Circles Modified by Lisa Palen. Definitions Circle The CENTER of the circle is the point that is the same distance to every point on the circle. The distance.

Circles Modified by Lisa Palen. Definitions Circle The CENTER of the circle is the point that is the same distance to every point on the circle. The distance.

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