+++
title = "Properties of circles"
description = "Apply chord, tangent, and angle theorems in circles to prove results and solve geometry problems."
template = "chapter.html"
weight = 28
[extra]
subject = "math"
chapter_id = "circle-properties"
strand = "Measures, Shape & Space"
+++

<div class="book-prose">
  <p>
    Circle theorems connect angles, chords, tangents, and arcs. They are essential for geometry proofs
    and exam reasoning.
  </p>

  <h2 id="key-theorems" class="book-prose__heading">Key circle theorems</h2>
  <ul>
    <li>Angle at center is twice angle at circumference on same arc.</li>
    <li>Angles in the same segment are equal.</li>
    <li>Angle in a semicircle is \(90^\circ\).</li>
    <li>Opposite angles of cyclic quadrilateral sum to \(180^\circ\).</li>
    <li>Tangent is perpendicular to radius at point of contact.</li>
    <li>Two tangents from same external point are equal in length.</li>
    <li>Alternate segment theorem: angle between tangent and chord equals angle in opposite arc.</li>
  </ul>

  <h2 id="proof-usage" class="book-prose__heading">Using theorems in proofs</h2>
  <ol>
    <li>Mark known equal angles/lengths clearly.</li>
    <li>Name theorem used at each step.</li>
    <li>Chain results to target statement.</li>
  </ol>

  <h2 id="cyclic-quadrilateral" class="book-prose__heading">Cyclic quadrilaterals</h2>
  <p>
    A quadrilateral is cyclic if all vertices lie on one circle.
  </p>
  <p>
    Useful tests:
  </p>
  <ul>
    <li>Opposite angles supplementary.</li>
    <li>Exterior angle equals opposite interior angle.</li>
  </ul>

  <h2 id="tangent-chord" class="book-prose__heading">Tangent-chord applications</h2>
  <p>
    When a tangent touches a chord endpoint, use alternate segment theorem to connect tangent angle
    with an inscribed angle subtending that chord.
  </p>

  <h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
  <ul>
    <li>Label all equal angles immediately.</li>
    <li>Write theorem names, not just numbers.</li>
    <li>Avoid assuming symmetry unless given or proved.</li>
    <li>Finish with a clear conclusion sentence.</li>
  </ul>
</div>

<div class="book-prose">
  <h2 id="history" class="book-prose__heading">History</h2>
  <p>
    Circle theorems were developed in classical geometry and refined through Euclidean proofs. They
    remain central in modern school geometry due to their structured reasoning style.
  </p>

  <h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
  <details class="book-question">
    <summary class="book-question__prompt">If central angle is \(100^\circ\), what is inscribed angle on same arc?</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(50^\circ\).</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">In a cyclic quadrilateral, one angle is \(72^\circ\). Find opposite angle.</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(108^\circ\).</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">What angle is formed by radius and tangent at point of contact?</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(90^\circ\).</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">Two tangents from external point \(P\) touch at \(A,B\). Compare \(PA\) and \(PB\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(PA=PB\).</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">Why does one theorem name per step improve proof quality?</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> It makes logical justification explicit and verifiable.</p>
    </div>
  </details>

  <h2 id="applications" class="book-prose__heading">Applications</h2>
  <ul>
    <li><strong>Design:</strong> arcs and tangency in mechanical/architectural drawings.</li>
    <li><strong>Optics:</strong> circular geometry in lenses and reflective paths.</li>
    <li><strong>CAD tools:</strong> tangent/chord constraints in modeling software.</li>
  </ul>

  <h2 id="references" class="book-prose__heading">References</h2>
  <ul>
    <li>Standard high-school geometry materials on circle theorems.</li>
    <li>Classical Euclidean geometry references.</li>
  </ul>
</div>