<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, sans-serif;
      background: #ffffff;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 10px;
      user-select: none;
    }

    .header {
      display: flex;
      align-items: center;
      justify-content: space-between;
      gap: 8px;
    }

    .badge-scale {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      padding: 4px 10px;
      border-radius: 20px;
      font-size: 13px;
      font-weight: 700;
      color: #0f172a;
    }

    .legend {
      display: flex;
      gap: 12px;
      font-size: 12px;
      font-weight: 600;
    }
    .legend-item {
      display: flex;
      align-items: center;
      gap: 5px;
    }
    .dot-navy { width: 10px; height: 10px; border-radius: 50%; background: #1e3a8a; }
    .dot-teal { width: 10px; height: 10px; border-radius: 50%; background: #0d9488; }

    .main-grid {
      display: grid;
      grid-template-columns: 1.15fr 1fr;
      gap: 12px;
      align-items: center;
    }
    @media (max-width: 480px) {
      .main-grid {
        grid-template-columns: 1fr;
        gap: 8px;
      }
    }

    /* Steps Column */
    .steps-col {
      display: flex;
      flex-direction: column;
      gap: 8px;
    }

    .step-card {
      background: #f8fafc;
      border: 1.5px solid #e2e8f0;
      border-radius: 8px;
      padding: 8px 10px;
      display: flex;
      align-items: center;
      justify-content: space-between;
      font-size: 13px;
      cursor: pointer;
      transition: background 0.2s, border-color 0.2s, box-shadow 0.2s;
    }
    .step-card.active {
      background: #eff6ff;
      border-color: #3b82f6;
      box-shadow: 0 0 0 1px #3b82f6;
    }

    .math-expr {
      font-family: ui-monospace, SFMono-Regular, Menlo, Monaco, Consolas, monospace;
      font-weight: 600;
      color: #334155;
      font-size: 12.5px;
    }
    .math-expr .orig { color: #1e3a8a; font-weight: 700; }
    .math-expr .factor { color: #d97706; font-weight: 700; }
    .math-expr .res { color: #0d9488; font-weight: 700; }

    /* Graph Column */
    .graph-container {
      width: 100%;
      aspect-ratio: 1.15 / 1;
      background: #fcfcfc;
      border-radius: 8px;
      border: 1px solid #e2e8f0;
      position: relative;
    }
    svg {
      width: 100%;
      height: 100%;
      display: block;
    }

    /* Controls */
    .controls {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 12px;
      margin-top: 2px;
    }
    .btn {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      border-radius: 6px;
      padding: 5px 12px;
      font-size: 12px;
      font-weight: 600;
      color: #334155;
      cursor: pointer;
      display: inline-flex;
      align-items: center;
      gap: 4px;
      transition: background 0.15s;
    }
    .btn:hover { background: #e2e8f0; }

    .dots-container {
      display: flex;
      gap: 6px;
    }
    .step-dot {
      width: 8px;
      height: 8px;
      border-radius: 50%;
      background: #cbd5e1;
      cursor: pointer;
      transition: background 0.2s, transform 0.2s;
    }
    .step-dot.active {
      background: #2563eb;
      transform: scale(1.25);
    }
  </style>
</head>
<body>

  <div class="header">
    <div class="badge-scale">Scale Factor: × 0.5 (from origin)</div>
    <div class="legend">
      <div class="legend-item"><span class="dot-navy"></span> PQR</div>
      <div class="legend-item"><span class="dot-teal"></span> P'Q'R'</div>
    </div>
  </div>

  <div class="main-grid">
    
    <div class="steps-col">
      <div class="step-card active" data-step="0">
        <div class="math-expr">
          <span class="orig">P(2, 4)</span> → (<span class="factor">0.5</span>·2, <span class="factor">0.5</span>·4) = <span class="res">P'(1, 2)</span>
        </div>
      </div>
      <div class="step-card" data-step="1">
        <div class="math-expr">
          <span class="orig">Q(6, 2)</span> → (<span class="factor">0.5</span>·6, <span class="factor">0.5</span>·2) = <span class="res">Q'(3, 1)</span>
        </div>
      </div>
      <div class="step-card" data-step="2">
        <div class="math-expr">
          <span class="orig">R(4, 0)</span> → (<span class="factor">0.5</span>·4, <span class="factor">0.5</span>·0) = <span class="res">R'(2, 0)</span>
        </div>
      </div>
    </div>

    
    <div class="graph-container">
      <svg viewBox="0 0 280 230" id="gridSvg">
        
        <g stroke="#e2e8f0" stroke-width="1">
          
          <line x1="30" y1="20" x2="30" y2="190" stroke="#94a3b8" stroke-width="1.5" />
          <line x1="64" y1="20" x2="64" y2="190" />
          <line x1="98" y1="20" x2="98" y2="190" />
          <line x1="132" y1="20" x2="132" y2="190" />
          <line x1="166" y1="20" x2="166" y2="190" />
          <line x1="200" y1="20" x2="200" y2="190" />
          <line x1="234" y1="20" x2="234" y2="190" />
          <line x1="268" y1="20" x2="268" y2="190" />

          
          <line x1="30" y1="190" x2="268" y2="190" stroke="#94a3b8" stroke-width="1.5" />
          <line x1="30" y1="147.5" x2="268" y2="147.5" />
          <line x1="30" y1="105" x2="268" y2="105" />
          <line x1="30" y1="62.5" x2="268" y2="62.5" />
          <line x1="30" y1="20" x2="268" y2="20" />
        </g>

        
        <g font-size="10" fill="#64748b" text-anchor="middle" font-family="monospace">
          <text x="30" y="204">0</text>
          <text x="64" y="204">1</text>
          <text x="98" y="204">2</text>
          <text x="132" y="204">3</text>
          <text x="166" y="204">4</text>
          <text x="200" y="204">5</text>
          <text x="234" y="204">6</text>
          <text x="268" y="204">7</text>

          <text x="20" y="151">1</text>
          <text x="20" y="109">2</text>
          <text x="20" y="66">3</text>
          <text x="20" y="24">4</text>
        </g>

        
        
        <line id="ray-0" x1="30" y1="190" x2="98" y2="20" stroke="#f59e0b" stroke-width="1.5" stroke-dasharray="4 3" opacity="0.4" />
        
        <line id="ray-1" x1="30" y1="190" x2="234" y2="105" stroke="#f59e0b" stroke-width="1.5" stroke-dasharray="4 3" opacity="0.4" />
        
        <line id="ray-2" x1="30" y1="190" x2="166" y2="190" stroke="#f59e0b" stroke-width="1.5" stroke-dasharray="4 3" opacity="0.4" />

        
        
        <polygon points="98,20 234,105 166,190" fill="#1e3a8a" fill-opacity="0.1" stroke="#1e3a8a" stroke-width="2" />
        
        
        <polygon points="64,105 132,147.5 98,190" fill="#0d9488" fill-opacity="0.25" stroke="#0d9488" stroke-width="2" />

        
        <circle cx="30" cy="190" r="3.5" fill="#475569" />
        <text x="24" y="185" font-size="10" font-weight="700" fill="#475569">O</text>

        
        <g id="v-p">
          <circle cx="98" cy="20" r="4" fill="#1e3a8a" />
          <text x="98" y="13" font-size="11" font-weight="700" fill="#1e3a8a" text-anchor="middle">P(2,4)</text>
        </g>
        <g id="v-q">
          <circle cx="234" cy="105" r="4" fill="#1e3a8a" />
          <text x="242" y="102" font-size="11" font-weight="700" fill="#1e3a8a">Q(6,2)</text>
        </g>
        <g id="v-r">
          <circle cx="166" cy="190" r="4" fill="#1e3a8a" />
          <text x="172" y="184" font-size="11" font-weight="700" fill="#1e3a8a">R(4,0)</text>
        </g>

        
        <g id="v-pp">
          <circle cx="64" cy="105" r="4" fill="#0d9488" />
          <text x="56" y="98" font-size="11" font-weight="700" fill="#0d9488" text-anchor="end">P'(1,2)</text>
        </g>
        <g id="v-qp">
          <circle cx="132" cy="147.5" r="4" fill="#0d9488" />
          <text x="140" y="144" font-size="11" font-weight="700" fill="#0d9488">Q'(3,1)</text>
        </g>
        <g id="v-rp">
          <circle cx="98" cy="190" r="4" fill="#0d9488" />
          <text x="98" y="182" font-size="11" font-weight="700" fill="#0d9488" text-anchor="middle">R'(2,0)</text>
        </g>

        
        <circle id="focus-orig" cx="98" cy="20" r="8" fill="none" stroke="#2563eb" stroke-width="2" opacity="0.9" />
        <circle id="focus-dil" cx="64" cy="105" r="8" fill="none" stroke="#0d9488" stroke-width="2" opacity="0.9" />
      </svg>
    </div>
  </div>

  
  <div class="controls">
    <button class="btn" id="prevBtn">◀ Prev</button>
    <div class="dots-container">
      <div class="step-dot active" data-step="0"></div>
      <div class="step-dot" data-step="1"></div>
      <div class="step-dot" data-step="2"></div>
    </div>
    <button class="btn" id="nextBtn">Next ▶</button>
    <button class="btn" id="playBtn" style="margin-left: 6px;">⏸</button>
  </div>

  <script src="{{VENDOR_CDN}}/classace-diagram.js"></script>
  <script>
    const coords = [
      { orig: [98, 20], dil: [64, 105] },      // P -> P'
      { orig: [234, 105], dil: [132, 147.5] }, // Q -> Q'
      { orig: [166, 190], dil: [98, 190] }     // R -> R'
    ];

    let currentStep = 0;
    let timer = null;
    let isPlaying = true;

    const cards = $$('.step-card');
    const dots = $$('.step-dot');
    const rays = [$('#ray-0'), $('#ray-1'), $('#ray-2')];
    const focusOrig = $('#focus-orig');
    const focusDil = $('#focus-dil');
    const playBtn = $('#playBtn');

    function updateStep(step) {
      currentStep = (step + 3) % 3;

      cards.forEach((c, idx) => {
        c.classList.toggle('active', idx === currentStep);
      });

      dots.forEach((d, idx) => {
        d.classList.toggle('active', idx === currentStep);
      });

      rays.forEach((r, idx) => {
        if (idx === currentStep) {
          r.setAttribute('stroke', '#2563eb');
          r.setAttribute('stroke-width', '2.5');
          r.setAttribute('opacity', '1');
        } else {
          r.setAttribute('stroke', '#94a3b8');
          r.setAttribute('stroke-width', '1.5');
          r.setAttribute('opacity', '0.35');
        }
      });

      const pt = coords[currentStep];
      focusOrig.setAttribute('cx', pt.orig[0]);
      focusOrig.setAttribute('cy', pt.orig[1]);
      focusDil.setAttribute('cx', pt.dil[0]);
      focusDil.setAttribute('cy', pt.dil[1]);
    }

    function startAuto() {
      if (timer) clearInterval(timer);
      isPlaying = true;
      playBtn.textContent = '⏸';
      timer = setInterval(() => {
        updateStep(currentStep + 1);
      }, 3500);
    }

    function stopAuto() {
      if (timer) clearInterval(timer);
      isPlaying = false;
      playBtn.textContent = '▶';
    }

    cards.forEach((card, idx) => {
      onTap(card, () => {
        stopAuto();
        updateStep(idx);
      });
    });

    dots.forEach((dot, idx) => {
      onTap(dot, () => {
        stopAuto();
        updateStep(idx);
      });
    });

    onTap($('#prevBtn'), () => {
      stopAuto();
      updateStep(currentStep - 1);
    });

    onTap($('#nextBtn'), () => {
      stopAuto();
      updateStep(currentStep + 1);
    });

    onTap(playBtn, () => {
      if (isPlaying) {
        stopAuto();
      } else {
        updateStep(currentStep + 1);
        startAuto();
      }
    });

    // Initialize
    updateStep(0);
    startAuto();
  </script>
</body>
</html>

Dilations and Scale Factors from the Origin

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #ffffff;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100vh;
      padding: 16px 12px;
    }

    .main-view {
      display: flex;
      flex-wrap: wrap;
      justify-content: center;
      align-items: flex-end;
      gap: 20px;
      width: 100%;
      max-width: 520px;
      margin: auto 0;
      padding-top: 10px;
    }

    .figure-panel {
      display: flex;
      flex-direction: column;
      align-items: center;
      text-align: center;
    }

    .figure-title {
      font-size: 13px;
      font-weight: 700;
      color: #64748b;
      margin-bottom: 24px;
      text-transform: uppercase;
      letter-spacing: 0.5px;
    }

    /* Original Rectangle (2 x 3) */
    .orig-box {
      width: 96px;
      height: 64px;
      background: #e0f2fe;
      border: 2.5px solid #0284c7;
      border-radius: 6px;
      display: flex;
      align-items: center;
      justify-content: center;
      position: relative;
      font-size: 13px;
      font-weight: 700;
      color: #0369a1;
      box-shadow: 0 2px 6px rgba(2, 132, 199, 0.15);
    }

    /* Dilated Rectangle (4 x 6) */
    .dilated-box {
      width: 192px;
      height: 128px;
      border: 2.5px solid #2563eb;
      border-radius: 8px;
      display: grid;
      grid-template-columns: 1fr 1fr;
      grid-template-rows: 1fr 1fr;
      gap: 4px;
      padding: 4px;
      background: #dbeafe;
      position: relative;
      box-shadow: 0 4px 12px rgba(37, 99, 235, 0.18);
    }

    .tile {
      background: #93c5fd;
      border: 1.5px dashed #2563eb;
      border-radius: 4px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      font-size: 12px;
      font-weight: 700;
      color: #1e40af;
      cursor: pointer;
      transition: background-color 0.2s ease, color 0.2s ease, border-color 0.2s ease, box-shadow 0.2s ease;
      user-select: none;
    }

    .tile:hover {
      filter: brightness(1.05);
    }

    .tile.active {
      background: #3b82f6;
      color: #ffffff;
      border-style: solid;
      box-shadow: inset 0 0 0 1px #1d4ed8;
    }

    .tile-num {
      font-size: 10px;
      opacity: 0.9;
    }

    /* Dimension labels */
    .dim-w {
      position: absolute;
      top: -20px;
      left: 0;
      width: 100%;
      text-align: center;
      font-size: 13px;
      font-weight: 700;
      color: #334155;
      line-height: 1;
      pointer-events: none;
    }

    .dim-h {
      position: absolute;
      right: -22px;
      top: 50%;
      transform: translateY(-50%);
      font-size: 13px;
      font-weight: 700;
      color: #334155;
      line-height: 1;
      pointer-events: none;
    }

    /* Summary Bar */
    .summary-card {
      width: 100%;
      max-width: 500px;
      display: flex;
      justify-content: space-around;
      align-items: center;
      background: #f8fafc;
      border: 1.5px solid #e2e8f0;
      border-radius: 10px;
      padding: 10px 14px;
      gap: 10px;
      margin-top: 14px;
    }

    .metric-badge {
      display: flex;
      flex-direction: column;
      align-items: center;
      text-align: center;
    }

    .metric-label {
      font-size: 11px;
      font-weight: 700;
      color: #64748b;
      text-transform: uppercase;
      letter-spacing: 0.4px;
      margin-bottom: 2px;
    }

    .metric-val {
      font-size: 15px;
      font-weight: 800;
      color: #0f172a;
    }

    .metric-val span {
      color: #2563eb;
    }

    .divider {
      width: 1px;
      height: 32px;
      background: #cbd5e1;
    }

    .scale-arrow-wrap {
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      margin-bottom: 48px;
    }

    .scale-arrow {
      font-size: 20px;
      color: #64748b;
      font-weight: 700;
      line-height: 1;
    }

    .scale-text {
      font-size: 11px;
      font-weight: 700;
      color: #2563eb;
      margin-top: 2px;
    }

    @media (max-width: 420px) {
      .main-view {
        gap: 12px;
      }
      .orig-box {
        width: 80px;
        height: 54px;
        font-size: 12px;
      }
      .dilated-box {
        width: 160px;
        height: 108px;
      }
      .tile {
        font-size: 11px;
      }
      .metric-val {
        font-size: 13px;
      }
      .figure-title {
        font-size: 12px;
        margin-bottom: 22px;
      }
    }
  </style>
</head>
<body>

  <div class="main-view">
    
    <div class="figure-panel">
      <div class="figure-title">Original (k = 1)</div>
      <div class="orig-box">
        <span class="dim-w">3</span>
        <span class="dim-h">2</span>
        Area = 6
      </div>
    </div>

    
    <div class="scale-arrow-wrap">
      <div class="scale-arrow">→</div>
      <div class="scale-text">k = 2</div>
    </div>

    
    <div class="figure-panel">
      <div class="figure-title">Dilated (k = 2)</div>
      <div class="dilated-box">
        <span class="dim-w">6</span>
        <span class="dim-h">4</span>
        <div class="tile active" data-index="1" title="Click to toggle highlight">
          <span>Area 6</span>
          <span class="tile-num">①</span>
        </div>
        <div class="tile active" data-index="2" title="Click to toggle highlight">
          <span>Area 6</span>
          <span class="tile-num">②</span>
        </div>
        <div class="tile active" data-index="3" title="Click to toggle highlight">
          <span>Area 6</span>
          <span class="tile-num">③</span>
        </div>
        <div class="tile active" data-index="4" title="Click to toggle highlight">
          <span>Area 6</span>
          <span class="tile-num">④</span>
        </div>
      </div>
    </div>
  </div>

  
  <div class="summary-card">
    <div class="metric-badge">
      <span class="metric-label">Side Lengths</span>
      <span class="metric-val">× 2</span>
    </div>
    <div class="divider"></div>
    <div class="metric-badge">
      <span class="metric-label">Total Area</span>
      <span class="metric-val">× 2² = <span>× 4</span> (24)</span>
    </div>
  </div>

  <script>
    // Tap tiles to toggle highlight
    const tiles = document.querySelectorAll('.tile');
    tiles.forEach(tile => {
      tile.addEventListener('click', () => {
        tile.classList.toggle('active');
      });
    });
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="1:1">
  <title>Dilation from Origin</title>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, sans-serif;
      color: #1e2945;
      background: #ffffff;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      padding: 12px;
      user-select: none;
    }
    .top-bar {
      display: flex;
      flex-wrap: wrap;
      align-items: center;
      justify-content: center;
      gap: 8px;
      margin-bottom: 6px;
      width: 100%;
    }
    .formula-badge {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      border-radius: 8px;
      padding: 4px 10px;
      font-size: 14px;
      font-weight: 600;
      letter-spacing: 0.2px;
    }
    .btn-group {
      display: flex;
      gap: 6px;
    }
    .toggle-btn {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      color: #1e2945;
      padding: 5px 12px;
      border-radius: 6px;
      font-size: 13px;
      font-weight: 600;
      cursor: pointer;
      transition: all 0.2s ease;
    }
    .toggle-btn.active {
      background: #ea580c;
      color: #ffffff;
      border-color: #ea580c;
    }
    .svg-wrap {
      width: 100%;
      max-width: 420px;
      aspect-ratio: 1 / 1;
      position: relative;
    }
    svg {
      width: 100%;
      height: 100%;
      overflow: visible;
    }
    .grid-line { stroke: #f1f5f9; stroke-width: 1; }
    .axis { stroke: #64748b; stroke-width: 1.5; }
    .ray { stroke: #94a3b8; stroke-width: 1.5; stroke-dasharray: 4 4; }
    .tri-orig { fill: rgba(37, 99, 235, 0.18); stroke: #2563eb; stroke-width: 2; }
    .tri-dilated { fill: rgba(234, 88, 12, 0.25); stroke: #ea580c; stroke-width: 2.5; transition: all 0.25s ease-out; }
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    .point-dilated { fill: #ea580c; transition: all 0.25s ease-out; }
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Conjuntos numéricos: de los naturales a los reales

Identificación de números irracionales en la vida cotidiana

Expansión decimal periódica y no periódica

Conversión de decimal periódico a fracción generatriz

Representación geométrica de raíces en la recta real

Densidad y completitud de los números reales

math

Estructura y clasificación de los números reales

Dilations with Centers Not at the Origin

Finding Scale Factor from Figures on a Coordinate Plane

Similarity Ratios and Scale Factors

Similarity as a Composition of Rigid Motions and Dilations

Angle-Angle (AA) Triangle Similarity Criterion

Side-Angle-Side (SAS) and Side-Side-Side (SSS) Triangle Similarity Criteria

Thales' Theorem and the Triangle Proportionality Theorem

Using Similarity in Right Triangles: Geometric Mean

Solving Proportions in Overlapping and Nested Similar Triangles

Applications of Triangle Similarity in Indirect Measurement

Dilations, Homothety, and Triangle Similarity

Solving Problems with Geometric Mean in Right Triangles

Similar Shapes and Proportional Corresponding Sides

Identifying Corresponding Sides of Similar Figures

Ratio of Corresponding Sides in Similar Shapes

Understanding Scale Factor

Similarity, Proportionality, and Scale Factor

线段的比的概念与单位的统一

成比例线段的概念与项的顺序

比例的基本性质：内项积等于外项积与变式

比例中项的定义与线段计算

黄金分割的定义与黄金比值 (√5-1)/2 ≈ 0.618

古希腊帕特农神庙、蒙娜丽莎与人体中的黄金分割

基本事实：两条直线被一组平行线所截，对应线段成比例

推论：平行于三角形一边的直线截其它两边所得对应线段成比例

平行线分线段成比例的判定应用

比例线段与平行截线定理的综合计算技巧

比例线段、黄金分割与平行线分线段成比例

Dilation as a Transformation

A Shape and Its Dilation on the Coordinate Plane

Comparing and Contrasting Attributes of Dilated Figures

Effect of Scale Factor on Coordinates

Algebraic Representation of Dilations

Dilations Centered at the Origin

Dilations from Centers Other Than the Origin

Applying Positive Rational Scale Factors

Dilations on the Coordinate Plane

生活中的相似现象与相似图形的概念

全等与相似的关系：全等是相似比为 1 的特殊相似

相似多边形的定义：各角对应相等且各边对应成比例

相似比的定义与前后图形顺序的关系

正多边形的相似性：任意两个边数相同的正多边形都相似

相似多边形的周长比等于相似比

相似多边形的面积比等于相似比的平方

地图比例尺与相似图形面积推算

相似多边形未知边长与未知角度的计算

相似多边形判定中的反例与警示（角相等或边成比例单条件不足）

图形的相似与相似多边形

Scale Factors Greater Than One (Enlargement)

Scale Factors Less Than One (Reduction)

Graphing Dilations on the Coordinate Plane

Verifying Similarity Using Coordinates

Effect of Dilation on Perimeter and Area

Translations, Reflections, and Rotations on the Coordinate Plane

Dilation and Similarity Word Problems

Dilations and Similarity Review

Analyzing Dilations and Other Transformations

相似三角形判定预备定理：平行于一边的直线截得的三角形与原三角形相似

相似三角形判定定理一：两角分别相等的两个三角形相似 (AA)

两角分别相等判定定理 (AA) 的严密证明

相似三角形判定定理二：两边成比例且夹角相等的两个三角形相似 (SAS)

两边成比例且夹角相等判定定理 (SAS) 的几何证明

相似三角形判定定理三：三边成比例的两个三角形相似 (SSS)

三边成比例判定定理 (SSS) 的几何证明

直角三角形相似的特殊判定：斜边与一直角边成比例 (HL 相似)

相似三角形判定方法的综合选择策略

相似判定中的常见错误：两边成比例与非夹角相等的反例剖析

相似三角形的判定定理与逻辑证明

相似三角形性质：对应高的比等于相似比

相似三角形性质：对应中线与对应角平分线的比等于相似比

相似三角形周长比等于相似比的严密证明与应用

相似三角形面积比等于相似比的平方的严密证明与计算

双垂直（母子三角形）模型与射影定理推导

一线三等角（K 字型）相似模型解析与应用

旋转相似与大小手拉手几何模型

相似三角形中的辅助线构造：做平行线与截长补短

相似三角形在圆中的应用：相交弦与切割线性质

相似三角形性质的综合几何代数化求解

相似三角形的性质定理与经典几何模型

位似图形的概念：位似中心与对应点共线特征

位似比的定义与图形放大或缩小

位似中心在图形外部的画法（同向与反向放缩）

位似中心在图形内部或顶点上的位似作图

直角坐标系中以原点为位似中心的坐标规律 (kx, ky) 与 (-kx, -ky)

在直角坐标系中利用坐标变换画位似图形

电影放映机、幻灯机与皮影戏中的位似原理

数码照片缩放与计算机图形学中的矢量位似变换

位似与相似、平移、旋转的综合关系辨析

位似作图与坐标变换综合演练与反思

图形的位似：概念、作图与坐标放缩

相似建模基本思想：从不可直达测量到相似三角形抽象

测量方法一：利用太阳光影子与同一时刻物高与影长成比例（例 81 基础）

测量方法二：利用标杆与人眼视线对齐测量高度（例 81 进阶）

测量方法三：利用平面镜光的反射角等于入射角测量高度

测量方法四：河流宽度与不可达两点间距离的相似测量

中国古代重差术：刘徽《海岛算经》中的相似望峰测高

测量误差来源分析与实验方案对比优化

现代科技应用：相机镜头成像与无人机航拍尺寸计算

校园实地测量综合实践项目：测量旗杆与教学楼高度

实地测量项目报告撰写、数据复盘与交流答辩

图形相似的实际应用与间接测量建模

解直角三角形的概念与边角五元素关系

基本类型一：已知两边（两直角边或一直角边一斜边）解直角三角形

基本类型二：已知一边一锐角解直角三角形

仰角与俯角问题：测量建筑物与无人机高度

方向角（方位角）问题：航海避礁与航空飞行定位

坡度与坡角问题：堤坝防洪与公路铁路工程坡比计算

化一般三角形为直角三角形：作高线（分割或补形）解题策略

双直角三角形（背靠背与母子型）列方程联合求解

跨学科应用：物理力的正交分解与日地月天文距离间接估算

图形的变化与相似全单元体系建构与高中几何衔接

解直角三角形及其在实际工程测量中的应用

Understanding Congruence of Shapes

Conditions for Triangle Congruence (SSS, SAS, ASA, RHS)

Identifying Congruent Triangles

Understanding Similarity of Shapes

Conditions for Triangle Similarity

Identifying Similar Triangles

Congruence and Similarity of Triangles

Congruence and Similarity of Quadrilaterals

Properties of Quadrilaterals Using Congruent Triangles

Solving Problems with Quadrilateral Properties

Transformations and Congruence

Transformations and Similarity

Quadrilaterals and Geometric Transformations

Describing Position in Three Dimensions

Using 3D Coordinate Systems

Using Digital Tools to Solve Spatial Problems

Designing Algorithms to Test Congruency and Similarity

Spatial Reasoning and Computational Geometry

关于 x 轴和 y 轴对称的点的坐标变化规律

关于原点对称（中心对称）的点的坐标变化规律

以坐标轴为对称轴的多边形对称图形绘制与顶点坐标变换

沿单一坐标轴方向平移时点的坐标变化规律（“左减右加，上加下减”）

多边形沿坐标轴方向平移一定距离后图形的顶点坐标确定

多边形依次沿两个坐标轴方向平移后的复合平移关系与顶点坐标

以原点为位似中心时坐标按倍数放缩与位似图形的生成

图形与坐标单元体系总结与尺规作图前瞻衔接

图形的运动与坐标变换规律

Conceptos de congruencia y figuras geométricas congruentes

Criterios de congruencia de triángulos: LAL, ALA, LLL y LLA

Demostraciones geométricas básicas con triángulos congruentes

Concepto de semejanza y razón de proporcionalidad

Criterios de semejanza de triángulos: AA, LLL y LAL

Teorema de Tales: segmentos proporcionales entre paralelas

Aplicaciones del teorema de Tales: medición indirecta de alturas

Congruencia y semejanza de triángulos con el teorema de Tales

Teorema de Pitágoras: demostración geométrica y algebraica

Cálculo de hipotenusa y catetos en triángulos rectángulos

Recíproco del teorema de Pitágoras y clasificación de triángulos

Resolución de problemas contextualizados con Pitágoras

Teoremas de las alturas y del cateto en el triángulo rectángulo

Razones trigonométricas en el triángulo rectángulo: seno, coseno y tangente

Cálculo de lados y ángulos usando trigonometría

Problemas de ángulos de elevación y de depresión

Geometría del triángulo rectángulo: Pitágoras, relaciones métricas y trigonometría

OGM.9.TRN.2

Aligned learning resources

Related diagrams