Skip to content

Circle Theorems

Definition

Circle theorems are statements about the geometric properties of circles, involving angles, chords, tangents, secants, and arcs. These theorems provide rules for calculating unknown angles and lengths within circle diagrams.

Prerequisites

A good understanding of the following is necessary:

  1. Basic Shapes & Spatial Awareness (Circles, lines, basic definitions)
  2. Angle Relationships (Types of angles, supplementary, congruent)
  3. Triangles (Properties, congruence, similarity)
  4. Coordinate Geometry (Optional, for some proofs/applications)

Learning Objectives

After mastering this topic, you should be able to:

  1. Define key terms: circle, radius, diameter, chord, tangent, secant, arc (minor/major), sector, segment, central angle, inscribed angle.
  2. Understand and apply theorems relating central angles and inscribed angles to their intercepted arcs.
  3. Understand and apply theorems about angles formed by tangents, chords, and secants.
  4. Understand and apply theorems about the properties of chords and tangents.
  5. Solve geometric problems involving finding angles and lengths in circles using these theorems.

Key Theorems (Examples)

(Note: There are many circle theorems; these are some common ones.)

  1. Central Angle Theorem: The measure of a central angle is equal to the measure of its intercepted arc.
  2. Inscribed Angle Theorem: The measure of an inscribed angle is half the measure of its intercepted arc.
    • Corollary: Inscribed angles that intercept the same arc are congruent.
    • Corollary: An angle inscribed in a semicircle is a right angle (90°).
  3. Tangent-Radius Theorem: A tangent line to a circle is perpendicular to the radius drawn to the point of tangency.
  4. Tangent Segments Theorem: Tangent segments drawn to a circle from the same external point are congruent.
  5. Angles Formed by Intersecting Chords: The measure of the angle formed by two intersecting chords inside a circle is half the sum of the measures of the intercepted arcs.
  6. Angles Formed by Tangents/Secants: Theorems exist for angles formed by two tangents, two secants, or a tangent and a secant intersecting outside the circle (angle measure is half the difference of the intercepted arcs).
  7. Congruent Chords Theorem: In the same circle or congruent circles, congruent chords have congruent arcs (and vice versa).
  8. Chord Perpendicular to Radius: If a radius (or diameter) is perpendicular to a chord, it bisects the chord and its arc.

Examples

Example 1: Inscribed Angle Theorem (High School)

Problem: In a circle, an inscribed angle intercepts an arc measuring 80°. What is the measure of the inscribed angle? Solution: The measure of the inscribed angle is half the measure of its intercepted arc. Angle = 80° / 2 = 40°.

Example 2: Angle in a Semicircle (High School)

Problem: Triangle ABC is inscribed in a circle such that AC is a diameter. What is the measure of angle ABC? Solution: An angle inscribed in a semicircle is a right angle. Therefore, angle ABC = 90°.

Example 3: Tangent-Radius Theorem (High School)

Problem: Line segment AB is tangent to circle O at point A. If OA (radius) = 5 and OB = 13, find the length of the tangent segment AB. Solution: The radius OA is perpendicular to the tangent AB, forming a right triangle OAB with the right angle at A. Use the Pythagorean theorem: OA² + AB² = OB². 5² + AB² = 13² 25 + AB² = 169 AB² = 144 AB = √144 = 12.

Example 4: Intersecting Chords (High School)

Problem: Two chords AC and BD intersect inside a circle at point E. Arc AB measures 50° and arc CD measures 70°. Find the measure of angle AEB. Solution: The measure of angle AEB is half the sum of the intercepted arcs AB and CD. Angle AEB = ½ (Arc AB + Arc CD) = ½ (50° + 70°) = ½ (120°) = 60°.

Common Misconceptions

  1. Confusing Central and Inscribed Angles: Forgetting that the inscribed angle is half the arc, while the central angle is equal to the arc.
  2. Incorrectly Identifying Intercepted Arcs: Especially for angles formed by tangents and secants.
  3. Assuming Lines are Tangents: Applying tangent theorems when a line might be a secant.
  4. Errors with Angle Formulas: Mixing up whether to add or subtract arcs for angles formed by intersecting lines (inside vs. outside the circle).

Applications in SAT

Circle theorem problems appear regularly on the SAT, often requiring integration of multiple geometric concepts:

  1. Finding Angle Measures: Using inscribed angle, central angle, tangent-chord angle theorems.
  2. Finding Arc Measures: Relating angles back to arcs.
  3. Problems with Tangents: Using tangent-radius and tangent segment properties.
  4. Combining Circle Properties with Triangles/Polygons: Solving for angles or sides in figures inscribed in or circumscribed about circles.
  5. Coordinate Geometry: Problems involving the equation of a circle combined with tangent lines or inscribed shapes.

Advanced Connections

Circle theorems are important in:

  1. Trigonometry - Deriving relationships like the Law of Sines, understanding angles in the unit circle.
  2. Euclidean Geometry: Proving more complex geometric relationships.
  3. Calculus: Concepts related to curvature and tangent lines.
  4. Physics: Optics (reflection/refraction in curved mirrors/lenses), circular motion.

Practice Problems

  1. Basic: A central angle in a circle measures 110°. What is the measure of its intercepted arc?
  2. Intermediate: Two tangents are drawn to a circle from an external point P. If one tangent segment has length 8 cm, what is the length of the other tangent segment from P?
  3. Advanced: An inscribed quadrilateral ABCD has arc AB = 70°, arc BC = 100°, arc CD = 110°. Find the measure of angle DAB.
  4. SAT-Level: In the circle with center O shown below (assume diagram with tangent PA and secant PBC), arc AC = 130° and arc AB = 50°. What is the measure of angle P?