Circles and Coordinate Geometry flashcards

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Area and inverse areaRadius, Diameter, Circumference, and Area: Area gives the radius squared, so undo with a square root.
When should I use radius and diameter?Radius, Diameter, Circumference, and Area: Convert a diameter to a radius.
What method supports area and inverse area?Radius, Diameter, Circumference, and Area: r=√(A/π); d=2√(A/π)
CircumferenceRadius, Diameter, Circumference, and Area: One diameter times π gives the full distance around.
Radius and diameterRadius, Diameter, Circumference, and Area: A diameter is two radii end to end.
What must I check in radius, diameter, circumference, and area?Radius, Diameter, Circumference, and Area: Radius is half the diameter; circumference is a length and area is in square units
What method supports circumference?Radius, Diameter, Circumference, and Area: C=2πr=πd
What method supports radius and diameter?Radius, Diameter, Circumference, and Area: d=2r
When should I use area and inverse area?Radius, Diameter, Circumference, and Area: Recover a radius or diameter from a circular area.
When should I use circumference?Radius, Diameter, Circumference, and Area: Compute boundary length from radius or diameter.
A diameter and a triangleCentral Angles, Inscribed Angles, and Arcs: An angle looking across a diameter is a right angle.
When should I use central angle and arc measure?Central Angles, Inscribed Angles, and Arcs: Relate a central angle to its intercepted minor arc.
What method supports a diameter and a triangle?Central Angles, Inscribed Angles, and Arcs: angle subtending a diameter=90°
Inscribed anglesCentral Angles, Inscribed Angles, and Arcs: A circle-edge angle sees half its far arc.
Central angle and arc measureCentral Angles, Inscribed Angles, and Arcs: A center angle measures the same fraction of the circle as its arc.
What must I check in central angles, inscribed angles, and arcs?Central Angles, Inscribed Angles, and Arcs: An inscribed angle is half its intercepted arc not containing the vertex; a central angle equals its arc measure in degrees
What method supports inscribed angles?Central Angles, Inscribed Angles, and Arcs: inscribed angle=(intercepted arc)/2
What method supports central angle and arc measure?Central Angles, Inscribed Angles, and Arcs: arc measure in degrees=central angle measure
When should I use a diameter and a triangle?Central Angles, Inscribed Angles, and Arcs: Use a semicircle to establish a right angle.
When should I use inscribed angles?Central Angles, Inscribed Angles, and Arcs: Double an inscribed angle to recover its intercepted arc.
Radian arc and sector formulasArc Length and Sector Area: Radians make arc length equal radius times angle.
When should I use arc length in degrees?Arc Length and Sector Area: Take a central-angle fraction of circumference.
What method supports radian arc and sector formulas?Arc Length and Sector Area: s=rθ; A=r²θ/2 when θ is in radians
Sector area in degreesArc Length and Sector Area: Use the wedge’s fraction of the whole disk.
Arc length in degreesArc Length and Sector Area: Take the same fraction of the circle’s boundary.
What must I check in arc length and sector area?Arc Length and Sector Area: Use degrees over 360 or radians over 2π consistently; arc length is linear and sector area is quadratic in radius
What method supports sector area in degrees?Arc Length and Sector Area: sector area=(θ/360)πr²
What method supports arc length in degrees?Arc Length and Sector Area: s=(θ/360)·2πr
When should I use radian arc and sector formulas?Arc Length and Sector Area: Use radian measure directly in arc-length and sector-area formulas.
When should I use sector area in degrees?Arc Length and Sector Area: Take a central-angle fraction of circle area.
External tangent lengthChords, Tangents, and Radius Relationships: Center-to-external distance is the hypotenuse, not the tangent length.
When should I use tangent and radius angle?Chords, Tangents, and Radius Relationships: Use a radius-tangent right angle.
What method supports external tangent length?Chords, Tangents, and Radius Relationships: PT=√(OP²−r²)
Chord distanceChords, Tangents, and Radius Relationships: Half the chord and the center distance are the right-triangle legs.
Tangent and radius angleChords, Tangents, and Radius Relationships: The radius meets its tangent at a right angle.
What must I check in chords, tangents, and radius relationships?Chords, Tangents, and Radius Relationships: A tangent is perpendicular to the radius at the tangency point; chord bisectors through the center are perpendicular
What method supports chord distance?Chords, Tangents, and Radius Relationships: (chord/2)²+d²=r²
What method supports tangent and radius angle?Chords, Tangents, and Radius Relationships: radius ⟂ tangent at the tangency point
When should I use external tangent length?Chords, Tangents, and Radius Relationships: Use a right triangle from center, tangency point, and external point.
When should I use chord distance?Chords, Tangents, and Radius Relationships: Use half a chord with a perpendicular center distance.
Point on a circle and unknown radiusEquations of Circles and Completing the Square: A point on the circle gives one full radius distance.
When should I use center and radius form?Equations of Circles and Completing the Square: Read a center and radius from a circle equation.
What method supports point on a circle and unknown radius?Equations of Circles and Completing the Square: r²=(x₀−h)²+(y₀−k)²
Complete two squaresEquations of Circles and Completing the Square: Complete one square for x and another for y.
Center and radius formEquations of Circles and Completing the Square: The squares show distance from the center.
What must I check in equations of circles and completing the square?Equations of Circles and Completing the Square: In standard circle form the right side is radius squared; a positive value gives a real nondegenerate circle
What method supports complete two squares?Equations of Circles and Completing the Square: x²−2hx+y²−2ky+C=0 ⇒ r²=h²+k²−C
What method supports center and radius form?Equations of Circles and Completing the Square: (x−h)²+(y−k)²=r²
When should I use point on a circle and unknown radius?Equations of Circles and Completing the Square: Use a given center-point distance to determine a circle constant.
When should I use complete two squares?Equations of Circles and Completing the Square: Convert a general circle equation to standard form.
Line through a centered circleCoordinate Problems Involving Circles and Lines: Replace y with the line formula and solve the shared-coordinate condition.
When should I use horizontal line intersections?Coordinate Problems Involving Circles and Lines: Substitute a fixed y-value into a circle equation.
What method supports line through a centered circle?Coordinate Problems Involving Circles and Lines: x²+(mx)²=r² ⇒ (1+m²)x²=r²
Tangency by center distanceCoordinate Problems Involving Circles and Lines: The nearest line-to-center distance decides whether it cuts, touches, or misses.
Horizontal line intersectionsCoordinate Problems Involving Circles and Lines: Fix y, then solve the remaining square in x.
What must I check in coordinate problems involving circles and lines?Coordinate Problems Involving Circles and Lines: Intersection coordinates must satisfy both equations; a tangent corresponds to exactly one shared point
What method supports tangency by center distance?Coordinate Problems Involving Circles and Lines: horizontal line distance=|c−k|
What method supports horizontal line intersections?Coordinate Problems Involving Circles and Lines: (x−h)²=r²−(c−k)²
When should I use line through a centered circle?Coordinate Problems Involving Circles and Lines: Use substitution for a nonhorizontal line.
When should I use tangency by center distance?Coordinate Problems Involving Circles and Lines: Compare line-to-center distance with radius.