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Circles
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Vocabulary
• Interior
• Exterior
• Chord
• Secant
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Tangents
• Tangent– Perpendicular to radius
• Example:
– 2 tangents from external point• Same measure
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Circles
• Congruent
• Concentric
• Tangent– Internally tangent– Externally tangent
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Arcs & Chords
• Arc – Minor Arc– Major Arc– Semi-Circle
• Finding Measure:
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Arcs
• Adjacent Arcs
• Congruent Arcs – 2 Arcs with the same measure– Central Angles– Chords
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Congruent Arcs
• Examples
• Find RT• Find mCD
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Radii & Chords
• If radius is perpendicular to chord– Bisects Chord & Arc
• A perpendicular bisector of chord is a radius
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Radii & Chords
• Example: Find QR
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Examples
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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
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Segments of Circles
• Segment of circle– Area of arc bounded by chord
• Finding Area of Segment
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Examples
• Sector:
• Segment:– segment RST
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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
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Inscribed Angles
• Inscribed angle– Vertex on circle – Sides contain chords
• Measure of inscribed angle = ½ measure of arc– m<E = ½(mDF)
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Inscribed Angles
• If inscribed angle arcs are congruent– Intercept same arc or congruent arcs– THEN: Inscribed angles are congruent
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Example
• Find m<DEC
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Inscribed Angle
• Inscribed angle subtends a semicircle if and only if the angle is a right angle
• Example:
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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
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Internal Angle
• Intersect inside circle– Measure of vertical angles is ½ sum of arcs
• Example: – Find m<PQT
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External Angle
•
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Examples
• Find x 1.
2. 3.
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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
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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
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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
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Examples with graph paper