Straight Lines
Coordinate Geometry of Polygons
Grade 11
Question:
<p>If an equilateral triangle has one vertex at the point <span class="math">(0, 0)</span> and another at <span class="math">(3, \sqrt{3})</span>, then the coordinates of the third vertex is</p>
<p>(a) <span class="math">(0, 2\sqrt{3})</span></p>
<p>(b) <span class="math">(0, -2\sqrt{3})</span></p>
<p>(c) <span class="math">(-1, 2\sqrt{3})</span></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: For an equilateral triangle, all three sides must be equal in length. Use the distance formula to find the side length from the two given vertices, then use the constraint that the third vertex is equidistant from both given vertices.
<p><strong>Step 1:</strong> Find the distance between A(0, 0) and B(3, √3).</p><p>Using the distance formula: AB = √[(3-0)² + (√3-0)²] = √[9 + 3] = √12 = 2√3</p><p><strong>Step 2:</strong> Let the third vertex be C(x, y). For an equilateral triangle, we need AC = BC = AB = 2√3.</p><p><strong>Step 3:</strong> From AC = 2√3: x² + y² = 12 ... (i)</p><p><strong>Step 4:</strong> From BC = 2√3: (x-3)² + (y-√3)² = 12</p><p>Expanding: x² - 6x + 9 + y² - 2√3·y + 3 = 12</p><p>x² + y² - 6x - 2√3·y + 12 = 12</p><p>x² + y² - 6x - 2√3·y = 0 ... (ii)</p><p><strong>Step 5:</strong> Substitute equation (i) into equation (ii):</p><p>12 - 6x - 2√3·y = 0</p><p>6x + 2√3·y = 12</p><p>3x + √3·y = 6 ... (iii)</p><p><strong>Step 6:</strong> From equation (iii): y = (6 - 3x)/√3 = (6 - 3x)√3/3 = 2√3 - √3·x</p><p><strong>Step 7:</strong> Substitute into equation (i):</p><p>x² + (2√3 - √3·x)² = 12</p><p>x² + 12 - 4√3·√3·x + 3x² = 12</p><p>x² + 12 - 12x + 3x² = 12</p><p>4x² - 12x = 0</p><p>4x(x - 3) = 0</p><p>So x = 0 or x = 3</p><p><strong>Step 8:</strong> When x = 0: y = 2√3 - 0 = 2√3, giving C(0, 2√3)</p><p>When x = 3: y = 2√3 - 3√3 = -√3, giving C(3, -√3)</p><p><strong>Step 9:</strong> Verify C(0, 2√3): AC = √[0 + 12] = 2√3 ✓ and BC = √[9 + (2√3 - √3)²] = √[9 + 3] = 2√3 ✓</p><p><strong>∴ Answer:</strong> a (The third vertex is at (0, 2√3))</p>
Correct Answer: a