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

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

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

The perpendicular bisector of a chord contains the center of a circle. One more Chord….

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

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

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

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

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.

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°

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

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

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

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

#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