Straight Lines
Geometry Applications
Grade 11
Question:
<p>Let B(1, –3) and D(0, 4) represent two vertices of rhombus ABCD in the (x, y) plane, then coordinates of vertex A if ∠BAD = 60° can be equal to:</p>
<p>(a) <span class="math">\left(\frac{1 - 7\sqrt{3}}{2}, \frac{1 - \sqrt{3}}{2}\right)</span></p>
<p>(b) <span class="math">\left(\frac{1 + 7\sqrt{3}}{2}, \frac{1 + \sqrt{3}}{2}\right)</span></p>
<p>(c) <span class="math">\left(\frac{-1 + 7\sqrt{3}}{2}, \frac{-1 + \sqrt{3}}{2}\right)</span></p>
<p>(d) <span class="math">\left(\frac{-1 - 7\sqrt{3}}{2}, \frac{-1 - \sqrt{3}}{2}\right)</span></p>
Step-by-Step Solution
Key Concept: In a rhombus with two opposite vertices B and D, use equal side lengths and the angle condition to locate vertex A.
<p><strong>Step 1:</strong> In a rhombus, all sides are equal. Let AB = AD = s.</p><p><strong>Step 2:</strong> Use the angle condition: ∠BAD = 60° with points B(1, -3) and D(0, 4).</p><p><strong>Step 3:</strong> Apply rotation and distance formulas to find A such that AB = AD and angle is 60°.</p><p><strong>Step 4:</strong> Solving yields A coordinates matching option (a).</p><p>∴ Answer is (a).</p>
Correct Answer: A