All about Circles: Definitions, Formulas, Theorems

• A circle represents a closed two-dimensional geometric shape.
• All points on a circle maintain an equal distance from the circle's center.
• Circles appear in mathematics, engineering, and daily life.

Key Terms

• Center: The equidistant point from all points on the circle.
• Radius (r): The distance measured from the center to any point on the circle.
• Diameter (d): The distance across the circle that passes through the center; it is equal to 2r.
• Circumference (C): The distance that encompasses the circle.
• Chord: A line segment connecting two points on the circle.
• Secant: A line intersecting the circle at two points.
• Tangent: A line touching the circle at one point.
• Arc: A fraction of the circle's circumference.
• Sector: The area enclosed by two radii and an arc.
• Segment: The area enclosed by a chord and an arc.

Equations and Formulas

• Circumference: C = 2πr, π (pi) ≈ 3.14159.
• Area: A = πr².
• Diameter: d = 2r.

Circle Theorems

• Angle in a Semicircle: An angle inscribed within a semicircle forms a right angle (90°).
• Angle at the Center Theorem: The angle produced by an arc at the center of the circle is twice the angle produced by the same arc at any point on the circle's remaining part.
• Angles in the Same Segment: Angles produced by the same arc in the same segment exhibit equality.
• Cyclic Quadrilateral Theorem: Opposite angles of a cyclic quadrilateral (where all vertices lie on the circle) are supplementary, summing to 180°.
• Tangent-Radius Theorem: A tangent to a circle forms a right angle with the radius at the point of tangency.
• Alternate Segment Theorem: The angle formed between a tangent and a chord equals the angle within the alternate segment.

Equation of a Circle in Coordinate Geometry

• Standard Form: (x - h)² + (y - k)² = r², (h, k) represents the center, and r is the radius.
• Center at Origin: When the center aligns with the origin (0, 0), the equation simplifies to x² + y² = r².

Properties of Tangents

• A tangent only contacts the circle at one point.
• The tangent is perpendicular to the radius at the contact point.
• Tangents from an External Point: Tangents drawn from an external point to a circle possess equal lengths.

Arcs and Sectors

• Arc Length: Defined as l = (θ/360) × 2πr, l represents the arc length, and θ is the central angle in degrees.
• Sector Area: Defined as A = (θ/360) × πr², A represents the sector area, and θ is the central angle in degrees.

Segments of a Circle

• Area of a Segment: Determined by subtracting the triangle's area (formed by the chord and radii) from the sector's area.
• Area of Segment = Area of Sector - Area of Triangle.

Working with Chords

• Perpendicular Bisector: A line from the circle’s center, perpendicular to the chord, bisects the chord.
• Equal Chords: Chords of equal length reside equidistant from the center.
• Intersecting Chords Theorem: If chords AB and CD intersect at point E inside the circle, then AE × EB = CE × ED.

Cyclic Quadrilaterals

• Definition: A quadrilateral whose vertices all lie on a circle's circumference.
• Opposite Angles: Opposite angles in a cyclic quadrilateral are supplementary, totaling 180°.

Applications

• Engineering: Essential for circular gears, wheels, and mechanical parts.
• Architecture: Foundations for arches, domes, and circular windows.
• Physics: Integral in circular motion, wave behavior, and optics.
• Everyday Life: Evident in wheels, clocks, coins, and numerous commonplace items.

Circle Theorems: In Detail

• Angle at the Center Theorem:
  • The angle that an arc forms at the circle's center is twice the angle formed by the same arc at any point on the circumference.
  • If ∠AOB is the center angle and ∠ACB is the circumference angle created by arc AB, then ∠AOB = 2 × ∠ACB.
• Angle in a Semicircle:
  • A specific instance of the Angle at the Center Theorem.
  • If AC denotes the circle's diameter, then the angle ∠ABC, where B is any point on the circumference, measures 90°.
• Angles in the Same Segment:
  • Angles created by the same arc within the same segment of a circle are equal.
  • If ∠ADB and ∠ACB are angles formed by arc AB in the same segment, then ∠ADB = ∠ACB.
• Cyclic Quadrilateral Theorem:
  • In a cyclic quadrilateral ABCD, the angles opposite each other are supplementary.
  • ∠A + ∠C = 180° and ∠B + ∠D = 180°.
• Tangent-Radius Theorem:
  • A radius drawn to the point of tangency forms a right angle with the tangent.
  • With OT as the radius and PT as the tangent at point T, ∠OTP = 90°.
• Alternate Segment Theorem:
  • The angle formed by a tangent and a chord at the contact point mirrors the angle in the alternate segment.
  • If PT denotes a tangent and AB is a chord at point A, the angle between the tangent and chord (∠BAT) equals the angle in the alternate segment (∠ACB).

Circle Equations: Advanced

• General Form:
  • Expressed as x² + y² + 2gx + 2fy + c = 0.
  • The circle's center is located at (-g, -f), and the radius is calculated as √(g² + f² - c).
• Parametric Form:
  • For a circle centered at (h, k) with radius r:
    • x = h + r cos(θ)
    • y = k + r sin(θ)
  • θ serves as a parameter.

Problem Solving Tips

• Always begin with a clear diagram to represent the problem visually.
• Accurately note all given lengths, angles, and relevant details.
• Identify and apply appropriate circle theorems relevant to the problem.
• Construct equations using established formulas and theorems.
• Check solutions thoroughly, ensuring they are logical within the problem context.

Common Mistakes

• Radius and Diameter Confusion: Always verify whether you are working with the radius or the diameter to avoid errors.
• Incorrect Theorem Application: Always ensure the right theorem is applied to the given situation.
• Errors in Calculation: Pay close attention to precision in calculating areas, circumferences, and other parameters.
• Omitting Units: Always present final answers complete with the correct units of measure.

Advanced Concepts

• Radical Axis: This represents the locus of points from which tangents to two circles are of equal length.
• Coaxal Circles: A system of circles where each pair possesses the same radical axis.
• Inscribed and Circumscribed Circles:
  • An inscribed circle touches each side of the polygon once.
  • A circumscribed circle passes through all vertices of the polygon.

Explore the properties of circles, including key terms like radius, diameter, and circumference. Learn about important formulas for calculating area and circumference. Understand circle theorems related to angles, chords and tangents.