openfreebooks-bookgenerator/content/math/coordinate-geometry-foundations/supplement.html

<div class="book-prose">
  <h2 id="history" class="book-prose__heading">History</h2>
  <p>
    Coordinate geometry was formalized by Rene Descartes and Pierre de Fermat. Their work connected
    algebraic equations to geometric curves, creating a major foundation for modern mathematics.
  </p>

  <h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
  <h3 id="distance-derivation" class="book-prose__heading">Distance formula (step by step)</h3>
  <p>Let \(A(x_1,y_1)\) and \(B(x_2,y_2)\).</p>
  <ol>
    <li>Construct a right triangle by drawing horizontal and vertical guide lines.</li>
    <li>Its horizontal leg is \(|x_2-x_1|\), and vertical leg is \(|y_2-y_1|\).</li>
    <li>By Pythagoras, \(AB^2=(x_2-x_1)^2+(y_2-y_1)^2\).</li>
    <li>Take the positive square root because distance is nonnegative.</li>
  </ol>
  <p class="book-formula">\[
    AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
  \]</p>

  <h3 id="midpoint-derivation" class="book-prose__heading">Midpoint formula (step by step)</h3>
  <p>
    A midpoint splits a segment into two equal parts in both directions. So the \(x\)-coordinate is
    halfway between \(x_1\) and \(x_2\), and the \(y\)-coordinate is halfway between \(y_1\) and \(y_2\).
  </p>
  <p class="book-formula">\[
    M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)
  \]</p>

  <h3 id="slope-derivation" class="book-prose__heading">Slope formula and line forms</h3>
  <p>
    Slope is rise over run. For points \(A(x_1,y_1)\) and \(B(x_2,y_2)\):
  </p>
  <p class="book-formula">\[
    m=\frac{y_2-y_1}{x_2-x_1}
  \]</p>
  <p>
    Rearranging the same idea for a generic point \((x,y)\) on the line through \((x_1,y_1)\)
    gives point-slope form:
  </p>
  <p class="book-formula">\[
    y-y_1=m(x-x_1)
  \]</p>
  <p>
    Expanding gives slope-intercept form \(y=mx+c\), where \(c\) is the \(y\)-intercept.
  </p>

  <h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
  <details class="book-question">
    <summary class="book-question__prompt">State the quadrant of \((-4,-7)\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> Quadrant III.</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">Find distance between \((1,2)\) and \((7,10)\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(\sqrt{6^2+8^2}=\sqrt{100}=10\).</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">Find midpoint of \((-3,5)\) and \((9,-1)\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(\left(3,2\right)\).</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">Find slope through \((2,-1)\) and \((6,7)\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(m=\frac{7-(-1)}{6-2}=\frac84=2\).</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">Line with slope \(3\) through \((1,4)\): find equation.</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(y-4=3(x-1)\Rightarrow y=3x+1\).</p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">Explain in one sentence why the distance formula has a square root.</summary>
    <div class="book-question__solution">
      <p>
        <strong>Answer:</strong> Pythagoras gives \(AB^2\), so we take the positive square root to get
        the actual distance \(AB\).
      </p>
    </div>
  </details>
  <details class="book-question">
    <summary class="book-question__prompt">Why is slope undefined for a vertical line?</summary>
    <div class="book-question__solution">
      <p>
        <strong>Answer:</strong> For a vertical line, \(\Delta x=0\), so
        \(m=\frac{\Delta y}{\Delta x}\) would require division by zero.
      </p>
    </div>
  </details>

  <h2 id="applications" class="book-prose__heading">Applications</h2>
  <ul>
    <li><strong>Maps/GPS:</strong> locations represented using coordinate systems.</li>
    <li><strong>Computer graphics:</strong> points and lines placed on pixel coordinate planes.</li>
    <li><strong>Physics:</strong> trajectories and motion graphs depend on coordinates and slope.</li>
  </ul>

  <h2 id="references" class="book-prose__heading">References</h2>
  <ul>
    <li>Standard high-school coordinate geometry units.</li>
    <li>Analytic geometry references on points, lines, distance, and midpoint.</li>
  </ul>
</div>