Circle Theorems: Foundations

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Video introduction to circle theorems

Essential Circle Theorems

Circle theorems describe the relationships between angles, radii, chords, and tangents in circles. These are fundamental for solving many SAT geometry problems.

Key Circle Theorems

Theorem Description Diagram
Central Angle Theorem The central angle is twice any inscribed angle subtending the same arc [Diagram showing central ∠AOB = 2× inscribed ∠ACB]
Inscribed Angle Theorem Inscribed angles subtending the same arc are equal [Diagram showing ∠ADB = ∠ACB]
Tangent-Radius Theorem A tangent to a circle is perpendicular to the radius at the point of contact [Diagram showing radius OP ⊥ tangent AB]
Chord Theorem Perpendicular from center to chord bisects the chord [Diagram showing OM ⊥ AB ⇒ AM = MB]
Cyclic Quadrilateral Opposite angles sum to 180° in a quadrilateral inscribed in a circle [Diagram showing ∠A + ∠C = 180°, ∠B + ∠D = 180°]

Important Circle Properties

Strategies for Solving Circle Problems

  1. Identify what's given: Recognize radii, chords, tangents, or angles
  2. Look for isosceles triangles: Two sides are often radii (equal length)
  3. Apply relevant theorems: Match the diagram to the theorems you know
  4. Mark equal angles/lengths: As you discover them, note them on the diagram
  5. Watch for right angles: Especially with tangents or perpendicular bisectors
  6. Consider supplementary angles: Common in cyclic quadrilaterals

Practice Question

[Diagram showing circle with center O, points A, B, C on circumference, angle AOB = 100°]

In the circle above with center O, points A, B, and C lie on the circumference. If angle AOB measures 100°, what is the measure of angle ACB?

A) 40°
B) 50°
C) 80°
D) 100°

Explanation

This problem requires applying the Central Angle Theorem, which states that the central angle (AOB) is twice the measure of any inscribed angle (ACB) that subtends the same arc (AB).

Given:

According to the Central Angle Theorem:

Central angle = 2 × Inscribed angle

Therefore:

100° = 2 × ∠ACB

∠ACB = 100° ÷ 2 = 50°

The correct answer is B) 50°.

Common mistakes:

Additional Tips