Advanced

Tutor model

i62 💰:9.4

i52 💰:7.9

🏆 i67 💰:62.5

i70 💰:125.0

i68 💰:125.0

🏆 i66 💰:75.0

i65 💰:125.0

i63 💰:62.5

i61 💰:75.0

i60 💰:9.4

i60 💰:37.5

i58 💰:62.5

i57 💰:22.5

i57 💰:37.5

i56 💰:45.0

⚡ i55 💰:11.25

i55 💰:37.5

i54 💰:62.5

i52 💰:35.0

i51 💰:37.5

i49 💰:15.0

i49 💰:62.5

i48 💰:11.3

i47 💰:7.5

i44 💰:25.0

i43 💰:37.5

i42 💰:187.5

i40 💰:5.0

i39 💰:100.0

i38 💰:37.5

i37 💰:11.0

i36 💰:187.5

i34 💰:12.5

i33 💰:37.5

i30 💰:1.0

i29 💰:1.5

i28 💰:150.0

i23 💰:20.0

i20 💰:4.0

i18 💰:1.0

i16 💰:25.0

i13 💰:1.5

Keep session private

What would you like to learn?

Override for the tutor's lesson content generation calls. Auto uses the system default (Claude Sonnet 5 for lesson text, Opus 5 for diagrams).

Lesson generation model

System default

i58 💰:18.8

i31 💰:11.0

i25 💰:31.3

Tutoring abilities

Imagine rolling a bicycle wheel along a flat, smooth floor.

Even though the wheel is round, it touches the flat floor at only **one single point** at any moment.

<!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">
  <title>Tangent-Radius Perpendicularity</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      user-select: none;
      -webkit-user-select: none;
    }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: transparent;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 8px 12px;
    }
    .diagram-wrap {
      width: 100%;
      max-width: 440px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 10px;
    }
    svg {
      width: 100%;
      height: auto;
      touch-action: none;
      overflow: visible;
    }
    .btn-row {
      display: flex;
      flex-wrap: wrap;
      justify-content: center;
      gap: 8px;
      width: 100%;
    }
    button {
      background: #f1f5f9;
      color: #334155;
      border: 1px solid #cbd5e1;
      border-radius: 20px;
      padding: 6px 14px;
      font-size: 14px;
      font-weight: 600;
      cursor: pointer;
      display: inline-flex;
      align-items: center;
      gap: 4px;
      transition: background 0.2s, color 0.2s, border-color 0.2s;
    }
    button:hover {
      background: #e2e8f0;
    }
    button.active {
      background: #22b7ff;
      color: #ffffff;
      border-color: #0284c7;
    }
    .drag-hint {
      font-size: 13px;
      color: #64748b;
      display: flex;
      align-items: center;
      gap: 4px;
    }
  </style>
</head>
<body>

<div class="diagram-wrap">
  <svg id="svg" viewBox="0 0 400 300" aria-label="Tangent-Radius Perpendicular Diagram">
    <defs>
      
      <filter id="shadow" x="-10%" y="-10%" width="120%" height="120%">
        <feDropShadow dx="0" dy="3" stdDeviation="4" flood-opacity="0.08" />
      </filter>
    </defs>

    
    <line id="tangentLine" x1="50" y1="240" x2="350" y2="240" stroke="#0ea5e9" stroke-width="4" stroke-linecap="round" />

    
    <circle id="wheel" cx="200" cy="150" r="90" fill="#f8fafc" stroke="#94a3b8" stroke-width="3" filter="url(#shadow)" />

    
    <g id="wheelSpokes" stroke="#cbd5e1" stroke-width="1.5" stroke-dasharray="3,3">
      <line id="spoke1" x1="200" y1="150" x2="200" y2="60" />
      <line id="spoke2" x1="200" y1="150" x2="110" y2="150" />
      <line id="spoke3" x1="200" y1="150" x2="290" y2="150" />
    </g>

    
    <line id="radiusLine" x1="200" y1="150" x2="200" y2="240" stroke="#f43f5e" stroke-width="3.5" stroke-linecap="round" />

    
    <path id="rightAngle" d="M 186 240 L 186 226 L 200 226" fill="none" stroke="#f43f5e" stroke-width="2" />
    <text id="angleText" x="180" y="222" font-size="13" font-weight="700" fill="#f43f5e" text-anchor="end">90°</text>

    
    <circle cx="200" cy="150" r="4.5" fill="#1e2945" />
    <text x="200" y="142" font-size="14" font-weight="700" fill="#1e2945" text-anchor="middle">O</text>

    
    <g id="contactGroup" style="cursor: grab;">
      <circle id="contactHalo" cx="200" cy="240" r="14" fill="#0ea5e9" fill-opacity="0.2" />
      <circle id="contactPoint" cx="200" cy="240" r="6" fill="#0284c7" stroke="#ffffff" stroke-width="2" />
    </g>

    
    <text id="labelRadius" x="212" y="195" font-size="13" font-weight="700" fill="#f43f5e">Radius</text>
    <text id="labelTangent" x="330" y="258" font-size="13" font-weight="700" fill="#0284c7" text-anchor="end">Tangent</text>
    <text id="labelPoint" x="200" y="272" font-size="13" font-weight="600" fill="#475569" text-anchor="middle">Point of contact</text>
  </svg>

  
  <div class="btn-row">
    <button class="active" onclick="setPreset(Math.PI / 2, this)">Floor</button>
    <button onclick="setPreset(Math.PI / 4, this)">Tilt 45°</button>
    <button onclick="setPreset(0, this)">Side</button>
    <button onclick="setPreset(-Math.PI / 2, this)">Top</button>
  </div>

  <div class="drag-hint">
    <span>✦</span> Drag point of contact around wheel
  </div>
</div>

<script>
  const svg = document.getElementById('svg');
  const tangentLine = document.getElementById('tangentLine');
  const radiusLine = document.getElementById('radiusLine');
  const rightAngle = document.getElementById('rightAngle');
  const angleText = document.getElementById('angleText');
  const contactHalo = document.getElementById('contactHalo');
  const contactPoint = document.getElementById('contactPoint');
  const labelRadius = document.getElementById('labelRadius');
  const labelTangent = document.getElementById('labelTangent');
  const labelPoint = document.getElementById('labelPoint');
  const spoke1 = document.getElementById('spoke1');
  const spoke2 = document.getElementById('spoke2');
  const spoke3 = document.getElementById('spoke3');

  const OX = 200, OY = 150, R = 90, TLEN = 135;
  let currentAngle = Math.PI / 2;
  let isDragging = false;

  function updateDiagram(theta) {
    currentAngle = theta;

    // Contact point P
    const px = OX + R * Math.cos(theta);
    const py = OY + R * Math.sin(theta);

    // Tangent unit direction
    const tx = -Math.sin(theta);
    const ty = Math.cos(theta);

    // Tangent line endpoints
    const x1 = px - TLEN * tx;
    const y1 = py - TLEN * ty;
    const x2 = px + TLEN * tx;
    const y2 = py + TLEN * ty;

    tangentLine.setAttribute('x1', x1);
    tangentLine.setAttribute('y1', y1);
    tangentLine.setAttribute('x2', x2);
    tangentLine.setAttribute('y2', y2);

    // Radius line
    radiusLine.setAttribute('x1', OX);
    radiusLine.setAttribute('y1', OY);
    radiusLine.setAttribute('x2', px);
    radiusLine.setAttribute('y2', py);

    // Contact point
    contactHalo.setAttribute('cx', px);
    contactHalo.setAttribute('cy', py);
    contactPoint.setAttribute('cx', px);
    contactPoint.setAttribute('cy', py);

    // Decorative secondary spokes
    const perpA = theta + Math.PI / 2;
    const oppA = theta + Math.PI;
    const perpA2 = theta - Math.PI / 2;
    spoke1.setAttribute('x1', OX); spoke1.setAttribute('y1', OY);
    spoke1.setAttribute('x2', OX + R * Math.cos(oppA)); spoke1.setAttribute('y2', OY + R * Math.sin(oppA));
    spoke2.setAttribute('x1', OX); spoke2.setAttribute('y1', OY);
    spoke2.setAttribute('x2', OX + R * Math.cos(perpA)); spoke2.setAttribute('y2', OY + R * Math.sin(perpA));
    spoke3.setAttribute('x1', OX); spoke3.setAttribute('y1', OY);
    spoke3.setAttribute('x2', OX + R * Math.cos(perpA2)); spoke3.setAttribute('y2', OY + R * Math.sin(perpA2));

    // Right angle square: size 14
    const s = 14;
    const urx = -Math.cos(theta);
    const ury = -Math.sin(theta);
    const utx = -Math.sin(theta);
    const uty = Math.cos(theta);

    const c1x = px + s * utx;
    const c1y = py + s * uty;
    const c2x = px + s * utx + s * urx;
    const c2y = py + s * uty + s * ury;
    const c3x = px + s * urx;
    const c3y = py + s * ury;

    rightAngle.setAttribute('d', `M ${c1x} ${c1y} L ${c2x} ${c2y} L ${c3x} ${c3y}`);

    // Angle label position
    const angLabelX = px + (s + 8) * utx + (s + 6) * urx;
    const angLabelY = py + (s + 8) * uty + (s + 6) * ury + 4;
    angleText.setAttribute('x', angLabelX);
    angleText.setAttribute('y', angLabelY);

    // Radius label position (midpoint + slight offset)
    const midX = (OX + px) / 2 - 12 * Math.sin(theta);
    const midY = (OY + py) / 2 + 12 * Math.cos(theta);
    labelRadius.setAttribute('x', midX);
    labelRadius.setAttribute('y', midY);

    // Tangent label position (near positive end of tangent)
    const tLabX = px + (TLEN - 25) * tx - 14 * Math.cos(theta);
    const tLabY = py + (TLEN - 25) * ty - 14 * Math.sin(theta);
    labelTangent.setAttribute('x', tLabX);
    labelTangent.setAttribute('y', tLabY);

    // Point of contact label position (outside wheel along radius)
    const pLabX = px + 24 * Math.cos(theta);
    const pLabY = py + 24 * Math.sin(theta) + 4;
    labelPoint.setAttribute('x', pLabX);
    labelPoint.setAttribute('y', pLabY);
  }

  function setPreset(theta, btn) {
    document.querySelectorAll('.btn-row button').forEach(b => b.classList.remove('active'));
    if (btn) btn.classList.add('active');
    updateDiagram(theta);
  }

  function getAngleFromEvent(e) {
    const rect = svg.getBoundingClientRect();
    const clientX = e.touches ? e.touches[0].clientX : e.clientX;
    const clientY = e.touches ? e.touches[0].clientY : e.clientY;
    const scaleX = 400 / rect.width;
    const scaleY = 300 / rect.height;
    const svgX = (clientX - rect.left) * scaleX;
    const svgY = (clientY - rect.top) * scaleY;
    return Math.atan2(svgY - OY, svgX - OX);
  }

  function onPointerDown(e) {
    isDragging = true;
    document.querySelectorAll('.btn-row button').forEach(b => b.classList.remove('active'));
    updateDiagram(getAngleFromEvent(e));
  }

  function onPointerMove(e) {
    if (!isDragging) return;
    e.preventDefault();
    updateDiagram(getAngleFromEvent(e));
  }

  function onPointerUp() {
    isDragging = false;
  }

  svg.addEventListener('mousedown', onPointerDown);
  window.addEventListener('mousemove', onPointerMove);
  window.addEventListener('mouseup', onPointerUp);

  svg.addEventListener('touchstart', onPointerDown, { passive: false });
  window.addEventListener('touchmove', onPointerMove, { passive: false });
  window.addEventListener('touchend', onPointerUp);

  // Initialize at floor (theta = 90 deg = PI / 2)
  updateDiagram(Math.PI / 2);
</script>

</body>
</html>

What special angle does that straight floor make with the wheel's spoke?

A **radius** is a straight line drawn from the center of a circle to its outside edge.

A **tangent** is a straight line outside the circle that grazes the circle at exactly one spot, called the **point of contact**.

<!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">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      user-select: none;
      -webkit-user-select: none;
    }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: transparent;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px;
    }
    .diagram-wrap {
      width: 100%;
      max-width: 420px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
    }
    svg {
      width: 100%;
      height: auto;
      touch-action: none;
      overflow: visible;
    }
    .handle {
      cursor: grab;
    }
    .handle:active {
      cursor: grabbing;
    }
    .btn-row {
      display: flex;
      gap: 8px;
      flex-wrap: wrap;
      justify-content: center;
    }
    .btn {
      background: #ffffff;
      border: 1px solid #e6e6e6;
      border-radius: 20px;
      padding: 6px 14px;
      font-size: 14px;
      font-weight: 500;
      color: #1e2945;
      cursor: pointer;
      box-shadow: 0 1px 3px rgba(0,0,0,0.05);
      transition: background 0.2s, border-color 0.2s;
    }
    .btn.active {
      background: #eaf7ff;
      border-color: #22b7ff;
      color: #0077b6;
      font-weight: 600;
    }
    .btn:hover {
      border-color: #22b7ff;
    }
  </style>
</head>
<body>
  <div class="diagram-wrap">
    <svg viewBox="0 0 400 360" id="svgCanvas">
      <defs>
        
        <filter id="drop-shadow" x="-30%" y="-30%" width="160%" height="160%">
          <feDropShadow dx="0" dy="2" stdDeviation="3" flood-color="#000" flood-opacity="0.2"/>
        </filter>
      </defs>

      
      <circle cx="200" cy="180" r="105" fill="#f8fafc" stroke="#cbd5e1" stroke-width="2.5" />
      
      
      <path id="rightAngle" fill="rgba(34, 183, 255, 0.15)" stroke="#0284c7" stroke-width="2" />
      
      
      <line id="tangentLine" stroke="#0ea5e9" stroke-width="3" stroke-linecap="round" />
      
      
      <line id="radiusLine" x1="200" y1="180" x2="200" y2="75" stroke="#64748b" stroke-width="2.5" stroke-dasharray="4 4" />
      
      
      <circle cx="200" cy="180" r="5" fill="#1e2945" />
      <text x="200" y="202" text-anchor="middle" font-size="14" font-weight="600" fill="#64748b">Center</text>

      
      <text id="radiusLabel" text-anchor="middle" dominant-baseline="middle" font-size="14" font-weight="600" fill="#475569">Radius</text>
      
      
      <text id="tangentLabel" text-anchor="middle" dominant-baseline="middle" font-size="14" font-weight="700" fill="#0284c7">Tangent</text>

      
      <text id="angleLabel" text-anchor="middle" dominant-baseline="middle" font-size="13" font-weight="700" fill="#0284c7">90°</text>

      
      <g id="contactGroup" class="handle">
        <circle id="contactPulse" cx="200" cy="75" r="18" fill="rgba(34, 183, 255, 0.2)" />
        <circle id="contactPoint" cx="200" cy="75" r="8" fill="#22b7ff" stroke="#ffffff" stroke-width="2.5" filter="url(#drop-shadow)" />
      </g>
      
      
      <text id="pointLabel" text-anchor="middle" font-size="14" font-weight="600" fill="#1e2945">Point of Contact</text>
    </svg>

    
    <div class="btn-row">
      <button class="btn active" onclick="setAngle(270, this)">Top</button>
      <button class="btn" onclick="setAngle(0, this)">Right</button>
      <button class="btn" onclick="setAngle(90, this)">Bottom</button>
      <button class="btn" onclick="setAngle(180, this)">Left</button>
      <button class="btn" id="playBtn" onclick="togglePlay()">✦ Rotate</button>
    </div>
  </div>

  <script>
    const CX = 200, CY = 180, R = 105, TAN_LEN = 110;
    let currentAngle = 270; // Start at top
    let isDragging = false;
    let isPlaying = false;
    let animFrame = null;

    const svg = document.getElementById('svgCanvas');
    const radiusLine = document.getElementById('radiusLine');
    const tangentLine = document.getElementById('tangentLine');
    const rightAngle = document.getElementById('rightAngle');
    const contactPoint = document.getElementById('contactPoint');
    const contactPulse = document.getElementById('contactPulse');
    const radiusLabel = document.getElementById('radiusLabel');
    const tangentLabel = document.getElementById('tangentLabel');
    const angleLabel = document.getElementById('angleLabel');
    const pointLabel = document.getElementById('pointLabel');
    const playBtn = document.getElementById('playBtn');

    function updateDiagram(deg) {
      currentAngle = (deg % 360 + 360) % 360;
      const rad = currentAngle * (Math.PI / 180);
      
      // Point of Contact P
      const px = CX + R * Math.cos(rad);
      const py = CY + R * Math.sin(rad);

      // Tangent vector
      const tx = -Math.sin(rad);
      const ty = Math.cos(rad);

      // Tangent line endpoints
      const t1x = px - tx * TAN_LEN;
      const t1y = py - ty * TAN_LEN;
      const t2x = px + tx * TAN_LEN;
      const t2y = py + ty * TAN_LEN;

      // Update Lines
      radiusLine.setAttribute('x2', px);
      radiusLine.setAttribute('y2', py);

      tangentLine.setAttribute('x1', t1x);
      tangentLine.setAttribute('y1', t1y);
      tangentLine.setAttribute('x2', t2x);
      tangentLine.setAttribute('y2', t2y);

      // Update Contact Point
      contactPoint.setAttribute('cx', px);
      contactPoint.setAttribute('cy', py);
      contactPulse.setAttribute('cx', px);
      contactPulse.setAttribute('cy', py);

      // 90° Corner Box (Size = 16)
      const s = 16;
      // Inward along radius:
      const rx = -Math.cos(rad);
      const ry = -Math.sin(rad);
      
      const pA_x = px + rx * s;
      const pA_y = py + ry * s;
      const pB_x = px + rx * s + tx * s;
      const pB_y = py + ry * s + ty * s;
      const pC_x = px + tx * s;
      const pC_y = py + ty * s;

      rightAngle.setAttribute('d', `M ${px} ${py} L ${pA_x} ${pA_y} L ${pB_x} ${pB_y} L ${pC_x} ${pC_y} Z`);

      // 90° Text Position
      const lblDist = 26;
      const angX = px + rx * lblDist + tx * (lblDist * 0.7);
      const angY = py + ry * lblDist + ty * (lblDist * 0.7);
      angleLabel.setAttribute('x', angX);
      angleLabel.setAttribute('y', angY);

      // Radius text (midpoint + slight offset)
      const midRx = CX + (R * 0.5) * Math.cos(rad) - ty * 16;
      const midRy = CY + (R * 0.5) * Math.sin(rad) + tx * 16;
      radiusLabel.setAttribute('x', midRx);
      radiusLabel.setAttribute('y', midRy);

      // Tangent text (near one end of tangent line)
      const tLabelX = px - tx * (TAN_LEN - 18) - rx * 16;
      const tLabelY = py - ty * (TAN_LEN - 18) - ry * 16;
      tangentLabel.setAttribute('x', tLabelX);
      tangentLabel.setAttribute('y', tLabelY);

      // Point of contact text (outside circle)
      const pLabelDist = 24;
      const pLblX = px + Math.cos(rad) * pLabelDist;
      const pLblY = py + Math.sin(rad) * pLabelDist + 5;
      pointLabel.setAttribute('x', pLblX);
      pointLabel.setAttribute('y', pLblY);
    }

    function setAngle(deg, btn) {
      if (isPlaying) togglePlay();
      document.querySelectorAll('.btn-row .btn').forEach(b => b.classList.remove('active'));
      if (btn) btn.classList.add('active');
      updateDiagram(deg);
    }

    function getAngleFromEvent(e) {
      const rect = svg.getBoundingClientRect();
      const clientX = e.touches ? e.touches[0].clientX : e.clientX;
      const clientY = e.touches ? e.touches[0].clientY : e.clientY;
      
      const svgX = (clientX - rect.left) * (400 / rect.width);
      const svgY = (clientY - rect.top) * (360 / rect.height);
      
      const dx = svgX - CX;
      const dy = svgY - CY;
      return Math.atan2(dy, dx) * (180 / Math.PI);
    }

    function startDrag(e) {
      isDragging = true;
      if (isPlaying) togglePlay();
      document.querySelectorAll('.btn-row .btn').forEach(b => b.classList.remove('active'));
      updateDiagram(getAngleFromEvent(e));
    }

    function onDrag(e) {
      if (!isDragging) return;
      e.preventDefault();
      updateDiagram(getAngleFromEvent(e));
    }

    function endDrag() {
      isDragging = false;
    }

    svg.addEventListener('mousedown', startDrag);
    window.addEventListener('mousemove', onDrag);
    window.addEventListener('mouseup', endDrag);

    svg.addEventListener('touchstart', startDrag, { passive: false });
    window.addEventListener('touchmove', onDrag, { passive: false });
    window.addEventListener('touchend', endDrag);

    function togglePlay() {
      isPlaying = !isPlaying;
      if (isPlaying) {
        playBtn.classList.add('active');
        playBtn.textContent = '⏸ Pause';
        let lastTime = performance.now();
        function loop(time) {
          if (!isPlaying) return;
          const dt = (time - lastTime) / 1000;
          lastTime = time;
          updateDiagram(currentAngle + dt * 45); // 45 deg/sec smooth rotation
          animFrame = requestAnimationFrame(loop);
        }
        animFrame = requestAnimationFrame(loop);
      } else {
        playBtn.classList.remove('active');
        playBtn.textContent = '✦ Rotate';
        cancelAnimationFrame(animFrame);
      }
    }

    // Initial draw
    updateDiagram(270);
  </script>
</body>
</html>

What happens right where the radius meets the tangent line?

When a radius meets a tangent at the point of contact, they make a perfect square corner, or a **right angle** ($90^\circ$).

We say the tangent is **perpendicular** to the radius, written as $\text{Radius} \perp \text{Tangent}$.

Over 2,300 years ago in ancient Greece, the mathematician Euclid recorded this fundamental rule in his famous geometry book, *Elements*.

<!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">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      user-select: none;
      -webkit-user-select: none;
    }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 8px;
    }
    .diagram-wrap {
      width: 100%;
      max-width: 420px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 10px;
    }
    .header-bar {
      display: flex;
      align-items: center;
      justify-content: space-between;
      width: 100%;
      padding: 0 8px;
    }
    .instruction {
      font-size: 13px;
      font-weight: 600;
      color: #6b7280;
      display: flex;
      align-items: center;
      gap: 6px;
    }
    .badge-90 {
      background: #e0f2fe;
      color: #0284c7;
      font-weight: 700;
      font-size: 14px;
      padding: 4px 10px;
      border-radius: 999px;
      box-shadow: 0 1px 2px rgba(0,0,0,0.05);
    }
    svg {
      width: 100%;
      height: auto;
      touch-action: none;
      cursor: crosshair;
      overflow: visible;
    }
    .drag-handle {
      cursor: grab;
      transition: filter 0.2s ease;
    }
    .drag-handle:active {
      cursor: grabbing;
      filter: drop-shadow(0 0 6px rgba(34, 183, 255, 0.8));
    }
    .presets {
      display: flex;
      gap: 8px;
      width: 100%;
      justify-content: center;
    }
    .preset-btn {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      color: #475569;
      font-size: 12px;
      font-weight: 600;
      padding: 6px 12px;
      border-radius: 8px;
      cursor: pointer;
      box-shadow: 0 1px 2px rgba(0,0,0,0.04);
      transition: background 0.15s ease, border-color 0.15s ease, color 0.15s ease;
    }
    .preset-btn:hover {
      background: #f1f5f9;
      border-color: #cbd5e1;
    }
    .preset-btn.active {
      background: #22b7ff;
      border-color: #22b7ff;
      color: #ffffff;
    }
  </style>
</head>
<body>

<div class="diagram-wrap">
  <div class="header-bar">
    <span class="instruction">✦ Drag point <strong>P</strong></span>
    <span class="badge-90" id="angleBadge">Angle = 90.0°</span>
  </div>

  <svg id="geoSvg" viewBox="0 0 400 400">
    
    <circle cx="200" cy="200" r="120" fill="#ffffff" stroke="#cbd5e1" stroke-width="2.5" stroke-dasharray="4 4" />
    
    
    <line id="tangentLine" x1="50" y1="80" x2="350" y2="80" stroke="#22b7ff" stroke-width="3.5" stroke-linecap="round" />
    
    
    <line id="radiusLine" x1="200" y1="200" x2="200" y2="80" stroke="#f59e0b" stroke-width="3" stroke-linecap="round" />
    
    
    <path id="rightAngle" d="" fill="none" stroke="#ef4444" stroke-width="2.5" stroke-linejoin="round" />
    
    
    <circle cx="200" cy="200" r="4.5" fill="#1e2945" />
    <text x="185" y="215" font-size="14" font-weight="700" fill="#64748b">O</text>

    
    <text id="radiusLabel" font-size="13" font-weight="600" fill="#b45309" text-anchor="middle">Radius</text>
    
    
    <text id="tangentLabel" font-size="13" font-weight="600" fill="#0284c7" text-anchor="middle">Tangent</text>

    
    <g id="handleGroup" class="drag-handle">
      <circle id="handleHalo" cx="200" cy="80" r="16" fill="rgba(34, 183, 255, 0.2)" />
      <circle id="handlePoint" cx="200" cy="80" r="7" fill="#22b7ff" stroke="#ffffff" stroke-width="2.5" />
      <text id="pointLabel" font-size="14" font-weight="700" fill="#1e2945">P</text>
    </g>
  </svg>

  <div class="presets">
    <button class="preset-btn active" data-angle="-90">Top</button>
    <button class="preset-btn" data-angle="0">Right</button>
    <button class="preset-btn" data-angle="90">Bottom</button>
    <button class="preset-btn" data-angle="180">Left</button>
  </div>
</div>

<script>
  const svg = document.getElementById('geoSvg');
  const tangentLine = document.getElementById('tangentLine');
  const radiusLine = document.getElementById('radiusLine');
  const rightAngle = document.getElementById('rightAngle');
  const radiusLabel = document.getElementById('radiusLabel');
  const tangentLabel = document.getElementById('tangentLabel');
  const handleGroup = document.getElementById('handleGroup');
  const handleHalo = document.getElementById('handleHalo');
  const handlePoint = document.getElementById('handlePoint');
  const pointLabel = document.getElementById('pointLabel');
  const presetBtns = document.querySelectorAll('.preset-btn');

  const CX = 200;
  const CY = 200;
  const R = 120;
  const TAN_LEN = 125;
  const SQ_SIZE = 16;

  let currentAngle = -Math.PI / 2; // top position
  let isDragging = false;

  function update(angleRad) {
    currentAngle = angleRad;

    // Contact point P
    const px = CX + R * Math.cos(angleRad);
    const py = CY + R * Math.sin(angleRad);

    // Tangent unit vector (perpendicular to radius)
    const tx = -Math.sin(angleRad);
    const ty = Math.cos(angleRad);

    // Tangent line end points
    const t1x = px - tx * TAN_LEN;
    const t1y = py - ty * TAN_LEN;
    const t2x = px + tx * TAN_LEN;
    const t2y = py + ty * TAN_LEN;

    tangentLine.setAttribute('x1', t1x);
    tangentLine.setAttribute('y1', t1y);
    tangentLine.setAttribute('x2', t2x);
    tangentLine.setAttribute('y2', t2y);

    // Radius line
    radiusLine.setAttribute('x1', CX);
    radiusLine.setAttribute('y1', CY);
    radiusLine.setAttribute('x2', px);
    radiusLine.setAttribute('y2', py);

    // Right angle square path inside the circle quadrant
    // Inward radius vector
    const rxIn = -Math.cos(angleRad);
    const ryIn = -Math.sin(angleRad);
    
    // Pick tangent direction for right-angle marker
    const p1x = px + rxIn * SQ_SIZE;
    const p1y = py + ryIn * SQ_SIZE;
    const p2x = px + rxIn * SQ_SIZE + tx * SQ_SIZE;
    const p2y = py + ryIn * SQ_SIZE + ty * SQ_SIZE;
    const p3x = px + tx * SQ_SIZE;
    const p3y = py + ty * SQ_SIZE;

    rightAngle.setAttribute('d', `M ${p1x} ${p1y} L ${p2x} ${p2y} L ${p3x} ${p3y}`);

    // Drag handle
    handleHalo.setAttribute('cx', px);
    handleHalo.setAttribute('cy', py);
    handlePoint.setAttribute('cx', px);
    handlePoint.setAttribute('cy', py);

    // P label position (shifted outward along radius)
    const labelDist = 26;
    const lx = px + Math.cos(angleRad) * labelDist;
    const ly = py + Math.sin(angleRad) * labelDist;
    pointLabel.setAttribute('x', lx - 5);
    pointLabel.setAttribute('y', ly + 5);

    // Radius label position (midpoint of radius, slightly offset)
    const midRx = (CX + px) / 2 - Math.sin(angleRad) * 14;
    const midRy = (CY + py) / 2 + Math.cos(angleRad) * 14;
    radiusLabel.setAttribute('x', midRx);
    radiusLabel.setAttribute('y', midRy + 4);

    // Tangent label position (near end of tangent line)
    const tanLabelX = t2x + tx * 6 + Math.cos(angleRad) * 14;
    const tanLabelY = t2y + ty * 6 + Math.sin(angleRad) * 14;
    tangentLabel.setAttribute('x', tanLabelX);
    tangentLabel.setAttribute('y', tanLabelY);
  }

  function getAngleFromEvent(e) {
    const rect = svg.getBoundingClientRect();
    const clientX = e.clientX || (e.touches && e.touches[0].clientX);
    const clientY = e.clientY || (e.touches && e.touches[0].clientY);
    const svgX = ((clientX - rect.left) / rect.width) * 400;
    const svgY = ((clientY - rect.top) / rect.height) * 400;
    return Math.atan2(svgY - CY, svgX - CX);
  }

  function clearActiveBtns() {
    presetBtns.forEach(btn => btn.classList.remove('active'));
  }

  svg.addEventListener('pointerdown', (e) => {
    isDragging = true;
    svg.setPointerCapture(e.pointerId);
    clearActiveBtns();
    update(getAngleFromEvent(e));
  });

  svg.addEventListener('pointermove', (e) => {
    if (!isDragging) return;
    update(getAngleFromEvent(e));
  });

  function stopDrag(e) {
    if (isDragging) {
      isDragging = false;
      try { svg.releasePointerCapture(e.pointerId); } catch (_) {}
    }
  }

  svg.addEventListener('pointerup', stopDrag);
  svg.addEventListener('pointercancel', stopDrag);

  presetBtns.forEach(btn => {
    btn.addEventListener('click', () => {
      clearActiveBtns();
      btn.classList.add('active');
      const deg = parseFloat(btn.dataset.angle);
      update((deg * Math.PI) / 180);
    });
  });

  // Initial draw
  update(currentAngle);
</script>

</body>
</html>

Why must this angle always be exactly $90^\circ$ instead of tilted?

The shortest path from a circle's center to a straight line is always a straight drop at a right angle.

Every other point on the tangent line sits outside the circle, making those paths longer than the radius.

<!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">
<title>Tangent-Radius Perpendicularity</title>
<style>
  * { box-sizing: border-box; margin: 0; padding: 0; }
  body {
    font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    color: #1e2945;
    background: transparent;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: center;
    min-height: 100%;
    padding: 12px;
    user-select: none;
  }
  .app-container {
    width: 100%;
    max-width: 440px;
    display: flex;
    flex-direction: column;
    align-items: center;
    gap: 12px;
  }
  .svg-wrap {
    width: 100%;
    aspect-ratio: 4 / 3;
    background: #ffffff;
    border-radius: 12px;
    box-shadow: 0 4px 16px rgba(0,0,0,0.06);
    overflow: hidden;
    position: relative;
    touch-action: none;
  }
  svg {
    width: 100%;
    height: 100%;
    display: block;
  }
  .controls {
    width: 100%;
    display: flex;
    flex-direction: column;
    gap: 8px;
    align-items: center;
  }
  .slider-row {
    width: 100%;
    display: flex;
    align-items: center;
    gap: 10px;
  }
  input[type=range] {
    flex: 1;
    height: 6px;
    -webkit-appearance: none;
    background: #e2e8f0;
    border-radius: 3px;
    outline: none;
  }
  input[type=range]::-webkit-slider-thumb {
    -webkit-appearance: none;
    width: 20px;
    height: 20px;
    border-radius: 50%;
    background: #22b7ff;
    cursor: pointer;
    box-shadow: 0 2px 6px rgba(34,183,255,0.4);
    border: 2px solid #ffffff;
  }
  .btn-group {
    display: flex;
    gap: 8px;
    width: 100%;
  }
  .btn {
    flex: 1;
    padding: 8px 12px;
    border: 1px solid #e2e8f0;
    background: #ffffff;
    color: #1e2945;
    font-size: 13px;
    font-weight: 600;
    border-radius: 8px;
    cursor: pointer;
    transition: all 0.2s ease;
    text-align: center;
  }
  .btn:hover {
    background: #f8fafc;
    border-color: #cbd5e1;
  }
  .btn.active {
    background: #22b7ff;
    color: #ffffff;
    border-color: #22b7ff;
    box-shadow: 0 2px 8px rgba(34,183,255,0.3);
  }
  .readout {
    font-size: 14px;
    font-weight: 600;
    display: flex;
    gap: 16px;
    color: #475569;
  }
  .highlight-val {
    color: #22b7ff;
  }
</style>
</head>
<body>

<div class="app-container">
  <div class="svg-wrap">
    <svg viewBox="0 0 400 300" id="diagram">
      
      <defs>
        <filter id="shadow" x="-10%" y="-10%" width="130%" height="130%">
          <feDropShadow dx="0" dy="2" stdDeviation="3" flood-opacity="0.1"/>
        </filter>
      </defs>

      
      <g stroke="#f1f5f9" stroke-width="1">
        <line x1="0" y1="60" x2="400" y2="60" />
        <line x1="0" y1="140" x2="400" y2="140" />
        <line x1="0" y1="220" x2="400" y2="220" />
        <line x1="120" y1="0" x2="120" y2="300" />
        <line x1="200" y1="0" x2="200" y2="300" />
        <line x1="280" y1="0" x2="280" y2="300" />
      </g>

      
      <circle cx="200" cy="140" r="80" fill="#f0f9ff" stroke="#22b7ff" stroke-width="2.5" />

      
      <line x1="30" y1="220" x2="370" y2="220" stroke="#94a3b8" stroke-width="2.5" stroke-dasharray="none" />
      <text x="375" y="224" font-size="12" fill="#64748b" font-weight="600">Tangent</text>

      
      <path id="rightAngle" d="M 200 206 L 214 206 L 214 220" fill="none" stroke="#22b7ff" stroke-width="2" />
      <text id="angleLabel" x="218" y="214" font-size="11" fill="#0284c7" font-weight="bold">90°</text>

      
      <polygon id="triangleFill" points="200,140 200,220 200,220" fill="rgba(34, 183, 255, 0.08)" />

      
      <line id="hypLine" x1="200" y1="140" x2="200" y2="220" stroke="#f59e0b" stroke-width="2.5" stroke-dasharray="4,4" />

      
      <line id="extLine" x1="200" y1="220" x2="200" y2="220" stroke="#ef4444" stroke-width="3" stroke-linecap="round" />

      
      <line x1="200" y1="140" x2="200" y2="220" stroke="#22b7ff" stroke-width="3" />

      
      <circle cx="200" cy="140" r="4.5" fill="#1e2945" />
      <text x="200" y="128" text-anchor="middle" font-size="13" font-weight="bold" fill="#1e2945">O (Center)</text>

      
      <text x="188" y="184" text-anchor="end" font-size="13" font-weight="bold" fill="#0284c7">r</text>

      
      <circle cx="200" cy="220" r="4.5" fill="#1e2945" />
      <text x="194" y="240" font-size="13" font-weight="bold" fill="#1e2945">P</text>

      
      <g id="pointQGroup" transform="translate(200, 220)">
        <circle id="pointQ" cx="0" cy="0" r="6" fill="#f59e0b" stroke="#ffffff" stroke-width="2" filter="url(#shadow)" style="cursor:ew-resize;" />
        <text id="labelQ" x="0" y="22" text-anchor="middle" font-size="13" font-weight="bold" fill="#b45309">Q</text>
      </g>

      
      <text id="hypLabel" x="250" y="170" font-size="12" font-weight="bold" fill="#d97706" text-anchor="middle" opacity="0">Hypotenuse > r</text>

      
      <rect x="20" y="16" width="360" height="28" rx="6" fill="#ffffff" stroke="#e2e8f0" />
      <text id="statusText" x="200" y="35" text-anchor="middle" font-size="13" font-weight="bold" fill="#1e2945">
        At contact point P: Shortest distance = Radius (90°)
      </text>
    </svg>
  </div>

  <div class="controls">
    <div class="slider-row">
      <span style="font-size:12px; font-weight:600; color:#64748b;">Position Q:</span>
      <input type="range" id="qSlider" min="-130" max="130" value="0" step="1">
    </div>
    <div class="btn-group">
      <button class="btn active" id="btnAtP">At Point P (90°)</button>
      <button class="btn" id="btnMoveQ">Move to Point Q</button>
    </div>
  </div>
</div>

<script>
  const slider = document.getElementById('qSlider');
  const pointQGroup = document.getElementById('pointQGroup');
  const hypLine = document.getElementById('hypLine');
  const extLine = document.getElementById('extLine');
  const triangleFill = document.getElementById('triangleFill');
  const rightAngle = document.getElementById('rightAngle');
  const angleLabel = document.getElementById('angleLabel');
  const hypLabel = document.getElementById('hypLabel');
  const statusText = document.getElementById('statusText');
  const btnAtP = document.getElementById('btnAtP');
  const btnMoveQ = document.getElementById('btnMoveQ');

  const Ox = 200, Oy = 140, R = 80, Py = 220;

  function update(dx) {
    const Qx = Ox + dx;
    const Qy = Py;

    pointQGroup.setAttribute('transform', `translate(${Qx}, ${Qy})`);
    hypLine.setAttribute('x2', Qx);
    hypLine.setAttribute('y2', Qy);

    triangleFill.setAttribute('points', `${Ox},${Oy} ${Ox},${Py} ${Qx},${Qy}`);

    if (Math.abs(dx) < 2) {
      // At contact point P
      extLine.setAttribute('x1', Ox);
      extLine.setAttribute('y1', Py);
      extLine.setAttribute('x2', Ox);
      extLine.setAttribute('y2', Py);
      hypLabel.setAttribute('opacity', '0');
      rightAngle.setAttribute('opacity', '1');
      angleLabel.setAttribute('opacity', '1');
      statusText.textContent = "At contact point P: Shortest distance = Radius (90°)";
      statusText.setAttribute('fill', '#0284c7');
      btnAtP.classList.add('active');
      btnMoveQ.classList.remove('active');
    } else {
      // Away from P: Hypotenuse > r
      const distOQ = Math.sqrt(dx * dx + R * R);
      const intersectX = Ox + (dx * R) / distOQ;
      const intersectY = Oy + (R * R) / distOQ;

      extLine.setAttribute('x1', intersectX);
      extLine.setAttribute('y1', intersectY);
      extLine.setAttribute('x2', Qx);
      extLine.setAttribute('y2', Qy);

      // Position Hypotenuse label near midpoint of OQ
      const midX = (Ox + Qx) / 2 + (dx > 0 ? 10 : -10);
      const midY = (Oy + Qy) / 2 - 8;
      hypLabel.setAttribute('x', midX);
      hypLabel.setAttribute('y', midY);
      hypLabel.setAttribute('opacity', '1');

      // Orient right angle symbol towards Q
      const sign = dx > 0 ? 1 : -1;
      rightAngle.setAttribute('d', `M ${Ox} ${Py - 14} L ${Ox + sign * 14} ${Py - 14} L ${Ox + sign * 14} ${Py}`);
      angleLabel.setAttribute('x', Ox + sign * 18);
      rightAngle.setAttribute('opacity', '1');
      angleLabel.setAttribute('opacity', '1');

      statusText.textContent = `Outside point Q: Path OQ > Radius (OQ = ${distOQ.toFixed(0)}px)`;
      statusText.setAttribute('fill', '#b45309');
      btnAtP.classList.remove('active');
      btnMoveQ.classList.add('active');
    }
  }

  slider.addEventListener('input', (e) => {
    update(parseFloat(e.target.value));
  });

  btnAtP.addEventListener('click', () => {
    slider.value = 0;
    update(0);
  });

  btnMoveQ.addEventListener('click', () => {
    slider.value = 95;
    update(95);
  });

  // Direct drag on SVG
  const svg = document.getElementById('diagram');
  let dragging = false;

  function handlePointer(e) {
    const rect = svg.getBoundingClientRect();
    const clientX = e.clientX || (e.touches && e.touches[0].clientX);
    if (clientX === undefined) return;
    const svgX = ((clientX - rect.left) / rect.width) * 400;
    const dx = Math.max(-130, Math.min(130, svgX - Ox));
    slider.value = dx;
    update(dx);
  }

  svg.addEventListener('pointerdown', (e) => {
    dragging = true;
    handlePointer(e);
  });
  window.addEventListener('pointermove', (e) => {
    if (dragging) handlePointer(e);
  });
  window.addEventListener('pointerup', () => { dragging = false; });

  // Init
  update(0);
</script>

</body>
</html>

How can we use this $90^\circ$ corner to find missing angles in a shape?

Look at triangle $\triangle OPT$ below, where $PT$ is a tangent line touching the circle at $P$, and the angle at the center $\angle POT = 58^\circ$.

<!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">
  <title>Tangent-Radius Perpendicularity</title>
  <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css">
  <script src="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.js"></script>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; font-family: system-ui, -apple-system, sans-serif; }
    body {
      background: #ffffff;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 10px 12px;
      user-select: none;
    }
    .main-svg {
      width: 100%;
      max-width: 440px;
      height: auto;
      display: block;
      overflow: visible;
    }
    .controls {
      display: flex;
      flex-wrap: wrap;
      align-items: center;
      justify-content: center;
      gap: 8px;
      margin-top: 6px;
      width: 100%;
      max-width: 440px;
    }
    .btn {
      background: #f1f5f9;
      border: 1.5px solid transparent;
      color: #1e2945;
      font-size: 13px;
      font-weight: 600;
      padding: 6px 12px;
      border-radius: 8px;
      cursor: pointer;
      transition: all 0.2s ease;
    }
    .btn.active {
      background: #e0f2fe;
      border-color: #22b7ff;
      color: #0284c7;
    }
    .calc-box {
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 6px 14px;
      font-size: 14px;
      font-weight: 500;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      min-height: 38px;
    }
    .label-badge {
      font-size: 12px;
      font-weight: 700;
      fill: #1e2945;
    }
  </style>
</head>
<body>

  <svg class="main-svg" viewBox="0 0 360 220">
    
    <circle cx="100" cy="140" r="75" fill="#f8fafc" stroke="#cbd5e1" stroke-width="2" />
    <circle cx="100" cy="140" r="3.5" fill="#1e2945" />

    
    <polygon id="tri-fill" points="100,140 100,65 220,65" fill="rgba(34, 183, 255, 0.08)" />

    
    <line id="tangent-ext" x1="45" y1="65" x2="310" y2="65" stroke="#22b7ff" stroke-width="2.5" stroke-dasharray="4,3" opacity="0.6" />
    <line id="tangent-seg" x1="100" y1="65" x2="220" y2="65" stroke="#22b7ff" stroke-width="3" stroke-linecap="round" />

    
    <line x1="100" y1="140" x2="100" y2="65" stroke="#0284c7" stroke-width="3" stroke-linecap="round" />

    
    <line id="hypot-line" x1="100" y1="140" x2="220" y2="65" stroke="#64748b" stroke-width="2" stroke-dasharray="3,3" />

    
    <path d="M 100,77 L 112,77 L 112,65" fill="none" stroke="#ef4444" stroke-width="2.5" />
    <rect x="104" y="69" width="3" height="3" fill="#ef4444" />

    
    <path id="center-arc" d="" fill="none" stroke="#0284c7" stroke-width="2" />
    
    
    <path id="t-arc" d="" fill="none" stroke="#64748b" stroke-width="2" />

    
    <circle id="point-p" cx="100" cy="65" r="4" fill="#0284c7" />
    <circle id="point-t" cx="220" cy="65" r="4" fill="#22b7ff" />

    
    <text x="82" y="152" class="label-badge">O</text>
    <text x="96" y="52" class="label-badge">P</text>
    <text id="label-t" x="225" y="58" class="label-badge">T</text>

    
    <text x="118" y="58" font-size="11" font-weight="700" fill="#ef4444">90°</text>
    <text id="val-o" x="114" y="128" font-size="12" font-weight="700" fill="#0284c7">58°</text>
    <text id="val-t" x="200" y="82" font-size="12" font-weight="700" fill="#475569">32°</text>

    
    <text x="56" y="106" font-size="11" font-weight="600" fill="#0284c7">Radius</text>
    <text x="250" y="58" font-size="11" font-weight="600" fill="#22b7ff">Tangent</text>
  </svg>

  <div class="calc-box" id="math-output"></div>

  <div class="controls">
    <button class="btn" onclick="setAngle(40)">∠POT = 40°</button>
    <button class="btn active" onclick="setAngle(58)">∠POT = 58°</button>
    <button class="btn" onclick="setAngle(65)">∠POT = 65°</button>
  </div>

  <script>
    const Ox = 100, Oy = 140, r = 75, Py = 65;

    function describeArc(cx, cy, radius, startAngleDeg, endAngleDeg) {
      const startRad = (startAngleDeg - 90) * Math.PI / 180;
      const endRad = (endAngleDeg - 90) * Math.PI / 180;
      const x1 = cx + radius * Math.cos(startRad);
      const y1 = cy + radius * Math.sin(startRad);
      const x2 = cx + radius * Math.cos(endRad);
      const y2 = cy + radius * Math.sin(endRad);
      return `M ${x1} ${y1} A ${radius} ${radius} 0 0 1 ${x2} ${y2}`;
    }

    function setAngle(degO) {
      // Update buttons
      document.querySelectorAll('.btn').forEach(btn => {
        btn.classList.toggle('active', btn.textContent.includes(degO + '°'));
      });

      const degT = 90 - degO;
      const radO = degO * Math.PI / 180;
      const Tx = Ox + r * Math.tan(radO);

      // Elements
      document.getElementById('tri-fill').setAttribute('points', `${Ox},${Oy} ${Ox},${Py} ${Tx},${Py}`);
      document.getElementById('tangent-seg').setAttribute('x2', Tx);
      document.getElementById('hypot-line').setAttribute('x2', Tx);
      document.getElementById('point-t').setAttribute('cx', Tx);
      document.getElementById('label-t').setAttribute('x', Tx + 6);

      // Dynamic Arc at O
      document.getElementById('center-arc').setAttribute('d', describeArc(Ox, Oy, 26, 0, degO));

      // Dynamic Arc at T
      const tArcRad = 24;
      const arcStartX = Tx - tArcRad;
      const arcStartY = Py;
      const angleAtTRad = Math.atan2(Oy - Py, Tx - Ox);
      const arcEndX = Tx - tArcRad * Math.cos(angleAtTRad);
      const arcEndY = Py + tArcRad * Math.sin(angleAtTRad);
      document.getElementById('t-arc').setAttribute('d', `M ${arcStartX} ${arcStartY} A ${tArcRad} ${tArcRad} 0 0 0 ${arcEndX} ${arcEndY}`);

      // Value text positions
      document.getElementById('val-o').textContent = `${degO}°`;
      document.getElementById('val-t').textContent = `${degT}°`;
      document.getElementById('val-t').setAttribute('x', Tx - 30);

      // Render KaTeX calculation
      const mathStr = `\\angle PTO = 180^\\circ - 90^\\circ - ${degO}^\\circ = \\mathbf{${degT}^\\circ}`;
      katex.render(mathStr, document.getElementById('math-output'), { throwOnError: false });
    }

    // Initialize with 58 degrees
    setAngle(58);
  </script>
</body>
</html>

Because $PT$ is a tangent, the angle at the contact point $\angle OPT$ is always $90^\circ$.

All three angles in any triangle add up to $180^\circ$, so we calculate: $\angle PTO = 180^\circ - 90^\circ - 58^\circ = 32^\circ$.

A line that cuts through the circle at two points is called a **secant**, not a tangent.

The $90^\circ$ rule only works when the line touches the circle at exactly one point.

<!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; user-select: none; -webkit-user-select: none; }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 12px;
      overflow: hidden;
    }
    .controls {
      display: flex;
      gap: 8px;
      z-index: 10;
      margin-bottom: 4px;
    }
    .btn {
      padding: 6px 14px;
      font-size: 13px;
      font-weight: 600;
      border-radius: 20px;
      border: 1px solid #d1d5db;
      background: #ffffff;
      color: #4b5563;
      cursor: pointer;
      box-shadow: 0 1px 2px rgba(0,0,0,0.05);
      transition: all 0.2s ease;
    }
    .btn.active {
      background: #22b7ff;
      color: #ffffff;
      border-color: #22b7ff;
      box-shadow: 0 2px 6px rgba(34, 183, 255, 0.35);
    }
    .stage {
      position: relative;
      width: 100%;
      max-width: 440px;
      flex: 1;
      display: flex;
      align-items: center;
      justify-content: center;
    }
    svg {
      width: 100%;
      height: 100%;
      max-height: 340px;
      touch-action: none;
    }
    .status-badge {
      display: flex;
      align-items: center;
      gap: 6px;
      font-size: 14px;
      font-weight: 700;
      background: #ffffff;
      padding: 6px 16px;
      border-radius: 20px;
      border: 1px solid #e5e7eb;
      box-shadow: 0 2px 4px rgba(0,0,0,0.04);
      margin-top: 4px;
    }
    .status-90 { color: #0284c7; }
    .status-not90 { color: #6b7280; }
  </style>
</head>
<body>

  <div class="controls">
    <button class="btn active" id="btnTangent">Tangent (1 Point)</button>
    <button class="btn" id="btnSecant">Secant (2 Points)</button>
  </div>

  <div class="stage">
    <svg id="svg" viewBox="-160 -150 320 300">
      <defs>
        <filter id="shadow" x="-20%" y="-20%" width="140%" height="140%">
          <feDropShadow dx="0" dy="1" stdDeviation="2" flood-opacity="0.15" />
        </filter>
      </defs>

      
      <circle cx="0" cy="0" r="90" fill="#ffffff" stroke="#cbd5e1" stroke-width="2.5" />
      
      
      <path id="angleMark" fill="rgba(34, 183, 255, 0.15)" stroke="#0284c7" stroke-width="1.5" />

      
      <line id="radiusLine" x1="0" y1="0" x2="0" y2="-90" stroke="#0284c7" stroke-width="3" stroke-linecap="round" />
      <circle cx="0" cy="0" r="4.5" fill="#1e2945" />
      <text x="-12" y="14" font-size="13" font-weight="700" fill="#64748b">O</text>
      
      
      <line id="tangentLine" x1="-150" y1="-90" x2="150" y2="-90" stroke="#1e2945" stroke-width="3" stroke-linecap="round" />

      
      <text id="radiusLabel" font-size="11" font-weight="750" fill="#0284c7" text-anchor="middle">Radius</text>
      
      
      <text id="lineLabel" font-size="11" font-weight="750" fill="#1e2945" text-anchor="middle">Tangent</text>

      
      <g id="handle" style="cursor: grab;">
        <circle id="handleGlow" cx="0" cy="-90" r="16" fill="rgba(34, 183, 255, 0.2)" />
        <circle id="handleDot" cx="0" cy="-90" r="7" fill="#0284c7" stroke="#ffffff" stroke-width="2" filter="url(#shadow)" />
        <text id="pointLabel" x="0" y="-104" font-size="13" font-weight="750" fill="#0284c7" text-anchor="middle">T</text>
      </g>

      
      <circle id="secantDot2" cx="0" cy="0" r="5" fill="#94a3b8" stroke="#ffffff" stroke-width="1.5" opacity="0" />
    </svg>
  </div>

  <div class="status-badge" id="statusBadge">
    <span id="badgeIcon">✦</span>
    <span id="badgeText" class="status-90">Radius ⊥ Tangent = 90°</span>
  </div>

  <script>
    const R = 90;
    let mode = 'tangent'; // 'tangent' or 'secant'
    let angle = -Math.PI / 2; // starting at top (0, -90)
    let isDragging = false;

    const svg = document.getElementById('svg');
    const radiusLine = document.getElementById('radiusLine');
    const tangentLine = document.getElementById('tangentLine');
    const radiusLabel = document.getElementById('radiusLabel');
    const lineLabel = document.getElementById('lineLabel');
    const angleMark = document.getElementById('angleMark');
    const handleGlow = document.getElementById('handleGlow');
    const handleDot = document.getElementById('handleDot');
    const pointLabel = document.getElementById('pointLabel');
    const secantDot2 = document.getElementById('secantDot2');
    const btnTangent = document.getElementById('btnTangent');
    const btnSecant = document.getElementById('btnSecant');
    const badgeText = document.getElementById('badgeText');
    const handle = document.getElementById('handle');

    function update() {
      const cosA = Math.cos(angle);
      const sinA = Math.sin(angle);
      
      if (mode === 'tangent') {
        const tx = R * cosA;
        const ty = R * sinA;

        // Radius
        radiusLine.setAttribute('x1', 0);
        radiusLine.setAttribute('y1', 0);
        radiusLine.setAttribute('x2', tx);
        radiusLine.setAttribute('y2', ty);
        radiusLine.setAttribute('stroke', '#0284c7');

        // Tangent direction is perpendicular to radius: (-sinA, cosA)
        const L = 120;
        const ux = -sinA;
        const uy = cosA;
        tangentLine.setAttribute('x1', tx - ux * L);
        tangentLine.setAttribute('y1', ty - uy * L);
        tangentLine.setAttribute('x2', tx + ux * L);
        tangentLine.setAttribute('y2', ty + uy * L);
        tangentLine.setAttribute('stroke', '#1e2945');

        // Handle position
        handleGlow.setAttribute('cx', tx);
        handleGlow.setAttribute('cy', ty);
        handleDot.setAttribute('cx', tx);
        handleDot.setAttribute('cy', ty);
        handleDot.setAttribute('fill', '#0284c7');

        // Label T outside
        pointLabel.setAttribute('x', tx + cosA * 18);
        pointLabel.setAttribute('y', ty + sinA * 18 + 4);
        pointLabel.textContent = 'T';

        // Radius label mid-line
        radiusLabel.setAttribute('x', (tx / 2) + uy * 12);
        radiusLabel.setAttribute('y', (ty / 2) - ux * 12 + 4);

        // Tangent line label
        lineLabel.setAttribute('x', tx + ux * 85 + cosA * 14);
        lineLabel.setAttribute('y', ty + uy * 85 + sinA * 14);
        lineLabel.textContent = 'Tangent';

        // Right angle square marker (size S)
        const S = 14;
        const rx = -cosA * S; // inward toward center along radius
        const ry = -sinA * S;
        const txNorm = ux * S; // along tangent
        const tyNorm = uy * S;
        
        const p1x = tx + txNorm;
        const p1y = ty + tyNorm;
        const p2x = tx + txNorm + rx;
        const p2y = ty + tyNorm + ry;
        const p3x = tx + rx;
        const p3y = ty + ry;

        angleMark.setAttribute('d', `M ${tx} ${ty} L ${p1x} ${p1y} L ${p2x} ${p2y} L ${p3x} ${p3y} Z`);
        angleMark.setAttribute('opacity', '1');
        angleMark.setAttribute('stroke', '#0284c7');
        angleMark.setAttribute('fill', 'rgba(34, 183, 255, 0.25)');

        secantDot2.setAttribute('opacity', '0');
        badgeText.className = 'status-90';
        badgeText.textContent = 'Radius ⊥ Tangent = 90°';
      } else {
        // Secant Mode: chord offset inside circle
        const d = 48; // distance of line from center
        const chordHalf = Math.sqrt(R * R - d * d);
        
        // Base line perpendicular to angle vector, offset by d
        const nx = cosA;
        const ny = sinA;
        const tx = -sinA;
        const ty = cosA;

        // Line passing through point (d*nx, d*ny) along direction (tx, ty)
        const midX = nx * d;
        const midY = ny * d;

        const L = 125;
        tangentLine.setAttribute('x1', midX - tx * L);
        tangentLine.setAttribute('y1', midY - ty * L);
        tangentLine.setAttribute('x2', midX + tx * L);
        tangentLine.setAttribute('y2', midY + ty * L);
        tangentLine.setAttribute('stroke', '#64748b');

        // Intersection points
        const i1x = midX + tx * chordHalf;
        const i1y = midY + ty * chordHalf;
        const i2x = midX - tx * chordHalf;
        const i2y = midY - ty * chordHalf;

        // Radius connects center to one intersection point
        radiusLine.setAttribute('x1', 0);
        radiusLine.setAttribute('y1', 0);
        radiusLine.setAttribute('x2', i1x);
        radiusLine.setAttribute('y2', i1y);
        radiusLine.setAttribute('stroke', '#94a3b8');

        // Handle at i1
        handleGlow.setAttribute('cx', i1x);
        handleGlow.setAttribute('cy', i1y);
        handleDot.setAttribute('cx', i1x);
        handleDot.setAttribute('cy', i1y);
        handleDot.setAttribute('fill', '#64748b');

        // Point label
        pointLabel.setAttribute('x', i1x + nx * 14 + tx * 8);
        pointLabel.setAttribute('y', i1y + ny * 14 + ty * 8 + 4);
        pointLabel.textContent = 'A';

        // 2nd dot
        secantDot2.setAttribute('cx', i2x);
        secantDot2.setAttribute('cy', i2y);
        secantDot2.setAttribute('opacity', '1');

        radiusLabel.setAttribute('x', (i1x / 2) - 10);
        radiusLabel.setAttribute('y', (i1y / 2) + 12);

        lineLabel.setAttribute('x', midX - tx * 85 + nx * 12);
        lineLabel.setAttribute('y', midY - ty * 85 + ny * 12);
        lineLabel.textContent = 'Secant';

        // Arc showing acute/obtuse angle (not 90°)
        const arcR = 20;
        const vRadX = -i1x / R;
        const vRadY = -i1y / R;
        const a1 = Math.atan2(vRadY, vRadX);
        const a2 = Math.atan2(-ty, -tx);
        
        const startX = i1x + arcR * Math.cos(a1);
        const startY = i1y + arcR * Math.sin(a1);
        const endX = i1x + arcR * Math.cos(a2);
        const endY = i1y + arcR * Math.sin(a2);

        angleMark.setAttribute('d', `M ${i1x} ${i1y} L ${startX} ${startY} A ${arcR} ${arcR} 0 0 0 ${endX} ${endY} Z`);
        angleMark.setAttribute('stroke', '#f59e0b');
        angleMark.setAttribute('fill', 'rgba(245, 158, 11, 0.2)');
        angleMark.setAttribute('opacity', '1');

        badgeText.className = 'status-not90';
        badgeText.textContent = '2 Points → Angle ≠ 90°';
      }
    }

    function setMode(m) {
      mode = m;
      if (mode === 'tangent') {
        btnTangent.classList.add('active');
        btnSecant.classList.remove('active');
      } else {
        btnSecant.classList.add('active');
        btnTangent.classList.remove('active');
      }
      update();
    }

    btnTangent.addEventListener('click', () => setMode('tangent'));
    btnSecant.addEventListener('click', () => setMode('secant'));

    function getAngleFromEvent(e) {
      const rect = svg.getBoundingClientRect();
      const clientX = e.touches ? e.touches[0].clientX : e.clientX;
      const clientY = e.touches ? e.touches[0].clientY : e.clientY;
      const x = clientX - (rect.left + rect.width / 2);
      const y = clientY - (rect.top + rect.height / 2);
      return Math.atan2(y, x);
    }

    function onPointerDown(e) {
      isDragging = true;
      handle.style.cursor = 'grabbing';
      angle = getAngleFromEvent(e);
      update();
    }

    function onPointerMove(e) {
      if (!isDragging) return;
      angle = getAngleFromEvent(e);
      update();
    }

    function onPointerUp() {
      isDragging = false;
      handle.style.cursor = 'grab';
    }

    svg.addEventListener('mousedown', onPointerDown);
    window.addEventListener('mousemove', onPointerMove);
    window.addEventListener('mouseup', onPointerUp);

    svg.addEventListener('touchstart', onPointerDown, { passive: true });
    window.addEventListener('touchmove', onPointerMove, { passive: true });
    window.addEventListener('touchend', onPointerUp);

    update();
  </script>
</body>
</html>

Whenever you spot a tangent line grazing a circle, look for the radius to instantly find your $90^\circ$ right angle.

math

Investigating the Tangent-Radius Perpendicularity Rule. Establish that a tangent to a circle is perpendicular to the radius at the point of contact.

Proof: Radius is Perpendicular to Tangent Line

Circle Geometry and Circle Theorems - Tangents and Radii

Tangent of a Circle: Step-by-Step Rules, Theorems, and Examples

Note on Level Classification: Standard MTH 4-17a is part of the Scottish Curriculum for Excellence Fourth Level (typically ages 12–15 / Secondary S1–S3). Pedagogical best practice for this standard progresses from concrete investigation (folding circular paper or using dynamic geometry software like GeoGebra to notice the invariant $90^\circ$ angle) to deductive reasoning (the shortest distance principle) and multi-step algebraic applications (Pythagorean theorem and trigonometry in right-angled triangles).

Investigating the Tangent-Radius Perpendicularity Rule

Naming the Parts

The Perpendicular Rule