mirror of
https://github.com/HananoshikaYomaru/openfreebooks.git
synced 2026-11-13 21:15:54 +00:00
69 lines
3 KiB
HTML
69 lines
3 KiB
HTML
<div class="book-prose">
|
|
<h2 id="history" class="book-prose__heading">History</h2>
|
|
<p>
|
|
Similarity ideas appear in ancient surveying and architecture. Greek geometers formalized triangle
|
|
relationships, enabling rigorous methods for unknown lengths and indirect measurements.
|
|
</p>
|
|
|
|
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
|
|
<h3 id="aa-why" class="book-prose__heading">Why AA implies triangle similarity</h3>
|
|
<p>
|
|
In triangles, angle sum is \(180^\circ\). If two angles match, the third angle must also match.
|
|
Matching angles force proportional corresponding sides, so triangles are similar.
|
|
</p>
|
|
|
|
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
|
<details class="book-question">
|
|
<summary class="book-question__prompt">
|
|
Triangles have side sets \((5,7,9)\) and \((5,7,9)\). Congruent or similar?
|
|
</summary>
|
|
<div class="book-question__solution">
|
|
<p><strong>Answer:</strong> Congruent (SSS, identical side lengths).</p>
|
|
</div>
|
|
</details>
|
|
<details class="book-question">
|
|
<summary class="book-question__prompt">
|
|
Two triangles have angles \(40^\circ,60^\circ,80^\circ\) and \(40^\circ,60^\circ,80^\circ\), but one is larger. What relation?
|
|
</summary>
|
|
<div class="book-question__solution">
|
|
<p><strong>Answer:</strong> Similar (AA), not necessarily congruent.</p>
|
|
</div>
|
|
</details>
|
|
<details class="book-question">
|
|
<summary class="book-question__prompt">
|
|
Similar triangles have scale factor \(k=3\). If a side is \(4\) in the small triangle, what is matching side in large triangle?
|
|
</summary>
|
|
<div class="book-question__solution">
|
|
<p><strong>Answer:</strong> \(4\times3=12\).</p>
|
|
</div>
|
|
</details>
|
|
<details class="book-question">
|
|
<summary class="book-question__prompt">
|
|
If two similar figures have area ratio \(25:9\), what is linear scale factor ratio?
|
|
</summary>
|
|
<div class="book-question__solution">
|
|
<p><strong>Answer:</strong> \(5:3\) (square root of area ratio).</p>
|
|
</div>
|
|
</details>
|
|
<details class="book-question">
|
|
<summary class="book-question__prompt">
|
|
Right triangles share one acute angle and each has a right angle. What criterion gives similarity?
|
|
</summary>
|
|
<div class="book-question__solution">
|
|
<p><strong>Answer:</strong> AA similarity.</p>
|
|
</div>
|
|
</details>
|
|
|
|
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
|
<ul>
|
|
<li><strong>Scale maps/drawings:</strong> convert plan dimensions to real distances.</li>
|
|
<li><strong>Shadow methods:</strong> estimate object heights using similar triangles.</li>
|
|
<li><strong>Engineering design:</strong> preserve shape while resizing components.</li>
|
|
</ul>
|
|
|
|
<h2 id="references" class="book-prose__heading">References</h2>
|
|
<ul>
|
|
<li>Standard high-school geometry syllabi on triangle congruence and similarity.</li>
|
|
<li>Geometry textbooks on ratio, scale factor, and proof applications.</li>
|
|
</ul>
|
|
</div>
|