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59 lines
2.6 KiB
HTML
59 lines
2.6 KiB
HTML
<div class="book-prose">
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<h2 id="history" class="book-prose__heading">History</h2>
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<p>
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Coordinate geometry was formalized by Rene Descartes and Pierre de Fermat. Their work connected
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algebraic equations to geometric curves, creating a major foundation for modern mathematics.
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</p>
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<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
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<h3 id="distance-derivation" class="book-prose__heading">Why the distance formula works</h3>
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<p>
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Horizontal change is \(\Delta x=x_2-x_1\), vertical change is \(\Delta y=y_2-y_1\). These form
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a right triangle, so distance is \(\sqrt{(\Delta x)^2+(\Delta y)^2}\).
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</p>
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<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
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<details class="book-question">
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<summary class="book-question__prompt">State the quadrant of \((-4,-7)\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> Quadrant III.</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Find distance between \((1,2)\) and \((7,10)\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(\sqrt{6^2+8^2}=\sqrt{100}=10\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Find midpoint of \((-3,5)\) and \((9,-1)\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(\left(3,2\right)\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Find slope through \((2,-1)\) and \((6,7)\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(m=\frac{7-(-1)}{6-2}=\frac84=2\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">Line with slope \(3\) through \((1,4)\): find equation.</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(y-4=3(x-1)\Rightarrow y=3x+1\).</p>
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</div>
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</details>
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<h2 id="applications" class="book-prose__heading">Applications</h2>
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<ul>
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<li><strong>Maps/GPS:</strong> locations represented using coordinate systems.</li>
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<li><strong>Computer graphics:</strong> points and lines placed on pixel coordinate planes.</li>
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<li><strong>Physics:</strong> trajectories and motion graphs depend on coordinates and slope.</li>
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</ul>
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<h2 id="references" class="book-prose__heading">References</h2>
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<ul>
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<li>Standard high-school coordinate geometry units.</li>
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<li>Analytic geometry references on points, lines, distance, and midpoint.</li>
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</ul>
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</div>
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