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

Equation for Circle (x – h) 2 + (y – k) 2 = r 2 – 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 (x 1 +x 2 )/2 Y coordinate is (y 1 +y 2 )/2 – Use center point coordinate and one end point with the distance formula to find the radius (√(x 1 -x c ) 2 +(y 1 -y c ) 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