Post on 22-Jun-2020
Chapter 9: Circles9.1 Circles and Chords
Vocabulary:Diameter
Radius
Chord
Central angle
Congruent circles
Objectives: Review sets of points associated with circles Identify and prove relationships and
theorems for congruent circles and chords.
Theorem: In a circle, if and only if a radius is perpendicular to a chord of a circle, then it bisects the chord.
Example:In the figure, circle P has a radius of 10. AB⊥DE .
If AB=8, what is the length of AC ?
If DC=2, what is the length of AB?
Theorem: In a circle or in congruent circles, if and only if two chords are the same distance from the center(s), the chords are congruent.
Example: Given circle A shown, AF=AG and BC=22. Find DE.
Chapter 9: Circles
CHALLENGE:The figure is a circle with center O and diameter 10 cm. PQ = 1 cm. Find the length of RS.
In this drawing, AB=30, OM=20, and ON=18. What is CN?
In the figure, AC ⊥BG, DF⊥EG, and EF=12. Find AC.
In the figure, CF=8, and the two concentric circles have radii of 10 and 17. Find DE.
9.2 TangentsVocabulary:Tangent
Secant
Point of tangency
Objectives:Identify and define tangents and secants.Prove theorems about tangents and secants.Classify common tangents and tangent circles.Use indirect proof
Chapter 9: CirclesTheorem: If and only if a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
Theorem: Tangent segments extending from a given external point to a circle are congruent.
Examples:In ⨀A, CB is tangent at point B. Find AC. Find the perimeter of △ABC.
Find the value of x. Determine whether the given segment is tangent to ⨀K.
9.3 ArcsObjectives: Identify and define relationships between arcs, central angles, and inscribed angles of circles. Identify minor arcs, major arcs, and semicircles. Prove theorems relating arcs, central angles, and chords in circles.Vocabulary:Central Angle:
Minor Arc
Major Arc
Semicircle
Chapter 9: Circles
ARC ADDITION POSTULATE
**total degrees around a circle are ______
Examples:
CONGRUENT CHORDS/ARCS
Examples:
Chapter 9: Circles
Prove the following:
9.4 Inscribed AnglesVocabularyInscribed angle
Objectives:Identify theorems relating inscribed angles to the measure of their intercepted arcs.
Theorem: The measure of an inscribed angle is one-half the measure of its intercepted arc.
Chapter 9: Circles
Examples:
Theorem: An angle inscribed in a semicircle is a right angle.
Example:
Theorem: The opposite angles of an inscribed quadrilateral are supplementary.Examples:
9.5 Lines and Circles
Objectives:Find the measures of angles formed by intersecting lines based on the measures of the intercepted arcs.
Chapter 9: Circles If the lines intersect OUTSIDE the circle:
Examples:
If the lines intersect INSIDE the circle:
Examples:
If the lines intersect ON the circle:
Chapter 9: Circles
Examples:
**do you notice a pattern in the formulas used for intersecting lines?
9.6 Sectors and SegmentsVocabulary:Sector of a circle
Segment of a circle
Objectives:Derive and apply a formula for arc length.Develop and apply formulas for the areas of sectors and segments.Develop and apply a formula for the perimeter of a sector.
Finding the length of an arc of a circle: L= πrθ180
Solve for the length of the following arcs:
Finding the area of a sector of a circle: A=A cθ360
Solve for the areas of the following sectors:
Chapter 9: Circles
Finding the area of a segment of a circle: A−A t
Solve for the area of the following segment:
Equations of CirclesObjectives:Graph a circle based off of its equation and vice versa.
General Equation of a Circle:
Graph the following circles:
x2+ y2=9 ( x−1 )2+( y+2 )2=16 ( x+5 )2+( y−4 )2=8
Write the equations for the following circles:
Chapter 9: Circles
9.7 Circle ConstructionsObjectives:Construct regular polygons inscribed in circles.
Construct the following:Inscribe a square in this circle: Inscribe an octagon in this circle: