openfreebooks-bookgenerator/content/math/pythagoras-theorem/supplement.html

<div class="book-prose">
  <h2 id="history" class="book-prose__heading">History</h2>
  <p>
    The theorem is associated with Pythagoras, but right-triangle relationships appeared in earlier
    Babylonian, Egyptian, and Indian mathematics. In Euclidean geometry, the theorem became a central
    tool for proof, measurement, and construction.
  </p>

  <h2 id="geometric-proof" class="book-prose__heading">Geometric proof by area</h2>
  <p>
    Build a large square of side length \(a+b\). Place four identical right triangles (legs \(a,b\),
    hypotenuse \(c\)) inside it so the uncovered center is a square of side \(c\).
  </p>
  <div class="math-widget-mount" data-widget="area-proof-demo" data-pagefind-ignore></div>
  <p>
    Area of big square: \((a+b)^2\). Area of the same figure as pieces:
    \(4\left(\frac{1}{2}ab\right)+c^2=2ab+c^2\). Set them equal:
  </p>
  <p class="book-formula">\[
    (a+b)^2 = 2ab + c^2
  \]</p>
  <p class="book-formula">\[
    a^2 + 2ab + b^2 = 2ab + c^2
  \]</p>
  <p class="book-formula">\[
    a^2 + b^2 = c^2
  \]</p>

  <h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
  <details class="book-question">
    <summary class="book-question__prompt">If the legs are 3 and 4, what is the hypotenuse?</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(c=\sqrt{3^2+4^2}=\sqrt{25}=5\).</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">If the legs are 6 and 8, what is the hypotenuse?</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(c=\sqrt{6^2+8^2}=\sqrt{100}=10\).</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">If the legs are 5 and 12, what is the hypotenuse?</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(c=\sqrt{5^2+12^2}=\sqrt{169}=13\).</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">A square has side length 6. Find its diagonal.</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(d=\sqrt{6^2+6^2}=\sqrt{72}=6\sqrt{2}\).</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">A square has side length 9. Find its diagonal.</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(d=\sqrt{9^2+9^2}=\sqrt{162}=9\sqrt{2}\).</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">If one leg is 4 and the hypotenuse is 8, find the other leg.</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(\sqrt{8^2-4^2}=\sqrt{64-16}=\sqrt{48}=4\sqrt{3}\).</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">If one leg is 10 and the hypotenuse is 15, find the other leg.</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(\sqrt{15^2-10^2}=\sqrt{225-100}=\sqrt{125}=5\sqrt{5}\).</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">For legs \(x\) and \(y\), write the hypotenuse.</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(c=\sqrt{x^2+y^2}\).</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">Using the proof figure, explain why \((a+b)^2\) and \(2ab+c^2\) must be equal.</summary>
    <div class="book-question__solution">
      <p>
        <strong>Answer:</strong> They are two area expressions for the same large square: one as a full
        square of side \(a+b\), and one as four right triangles plus the center square of side \(c\).
      </p>
    </div>
  </details>

  <h2 id="vocabulary" class="book-prose__heading">Vocabulary</h2>
  <table>
    <thead>
      <tr>
        <th>Term</th>
        <th>Definition</th>
      </tr>
    </thead>
    <tbody>
      <tr>
        <td><strong>Right triangle</strong></td>
        <td>A triangle with one angle equal to \(90^\circ\).</td>
      </tr>
      <tr>
        <td><strong>Legs</strong></td>
        <td>The two sides that form the right angle.</td>
      </tr>
      <tr>
        <td><strong>Hypotenuse</strong></td>
        <td>The side opposite the right angle; the longest side in a right triangle.</td>
      </tr>
      <tr>
        <td><strong>Pythagorean Theorem</strong></td>
        <td>The relationship \(a^2+b^2=c^2\) for right triangles.</td>
      </tr>
    </tbody>
  </table>
</div>