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: