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1 By Mark Hatem and Maddie Hines Now what you’ve all been waiting for..Chapter 12:CirclesBy Mark Hatem and Maddie Hines
2 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 equalA 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
4 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 chordChords in Lesson 1
5 The perpendicular bisector of a chord contains the center of a circle. One more Chord….
6 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
7 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 = mABLesson 3 – Angles and Arcs
8 Chords in Lesson 3 In a circle equal chords have equal arcs. In a circle equal arcs have equal chords.
9 An inscribed angle is an angle whose vertex is on a circle, with each of the angle’s sides intersecting the circle in another pointTheorem 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
10 Lesson 5 - SecantsA 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.
11 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=70120°10°
12 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
13 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
14 Angles and Arcs in Circles Notes Central Anglex°=a°Angle inscribed in a semicirclex°=90°Inscribed anglex°=1/2a°Angles inscribed in the same arcx°=y°=1/2a°Tangent and radius angleTangent and chord angleaxxaaxxyaxx
15 Angles and Arcs in Circles Notes Cnt. Two ChordsAnglex°=a°+b°2Two tangentsx°=(b-a)/2Or x°=180-aOr x°=180-bTwo Secantsx°= b-aSecant and TangentabxxabbbxaaX
16 #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 tangentsWhat's Important