Circles

Vocabulary

• Interior

• Exterior

• Chord

• Secant

Tangents

• Tangent– Perpendicular to radius

• Example:

– 2 tangents from external point• Same measure

Circles

• Congruent

• Concentric

• Tangent– Internally tangent– Externally tangent

Arcs & Chords

• Arc – Minor Arc– Major Arc– Semi-Circle

• Finding Measure:

Arcs

• Adjacent Arcs

• Congruent Arcs – 2 Arcs with the same measure– Central Angles– Chords

Congruent Arcs

• Examples

• Find RT• Find mCD

Radii & Chords

• If radius is perpendicular to chord– Bisects Chord & Arc

• A perpendicular bisector of chord is a radius

Radii & Chords

• Example: Find QR

Examples

Sectors & Arc Length

• Sector of a circle – 2 radii & arc– The pie shaped slice of the circle

• Area of sector is percent area of circle based on arc or central angle:

A= Area, r= radius, m= measure of arc/angle

Segments of Circles

• Segment of circle– Area of arc bounded by chord

• Finding Area of Segment

Examples

• Sector:

• Segment:– segment RST

Arc Length

• Distance along the arc (circumference)– Measured in linear units

– L= length, r= radius, m= measure of arc/angle

• Example:– Measure of GH

file://localhost/Users/cmidthun/Downloads/practice_a (39).doc

Inscribed Angles

• Inscribed angle– Vertex on circle – Sides contain chords

• Measure of inscribed angle = ½ measure of arc– m<E = ½(mDF)

Inscribed Angles

• If inscribed angle arcs are congruent– Intercept same arc or congruent arcs– THEN: Inscribed angles are congruent

Example

• Find m<DEC

Inscribed Angle

• Inscribed angle subtends a semicircle if and only if the angle is a right angle

• Example:

Angle Relationships

• Tangent and a secant/chord– Measure of angle is ½ intercepted arc measure– Measure of the arc is twice the measure of angle

• Example:– Find m<BCD– Find measure of arc AB

Internal Angle

• Intersect inside circle– Measure of vertical angles is ½ sum of arcs

• Example: – Find m<PQT

External Angle

•

Examples

• Find x 1.

2. 3.

Equation for Circle

• (x – h)2 + (y – k)2 = r2

– h is the x coordinate of the center point– k is the y coordinate of the center point– r is the radius

Finding center & radius

• Given 2 endpoints – Find center point

• X coordinate is (x1+x2)/2

• Y coordinate is (y1+y2)/2

– Use center point coordinate and one end point with the distance formula to find the radius• (√(x1-xc)2+(y1-yc)2 )

• Plug center point and radius into equation for circle

Slope of Tangent

• Slope of radius = Rise over Run (ΔY ÷ ΔX) • Find negative reciprocal – Change sign, flip fraction

• Insert negative reciprocal into slope formula– Y = mx + b– Substitute y & x coords from tangent point to find b

• Rewrite equation with y & x and the b value

Examples with graph paper