<p>A square, of each side 2, lies above the <em>x</em>-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle 30° with the positive direction of the <em>x</em>-axis, then the sum of the <em>x</em>-coordinates of the vertices of the square is</p>
Step-by-Step Solution
Key Concept: A square has two perpendicular sides meeting at the origin. If one side makes 30° with the x-axis, the other makes 120°. Use unit vectors along these directions scaled by side length 2 to find all four vertices, then sum their x-coordinates.
<p><strong>Step 1:</strong> One side from origin makes 30° with x-axis. The perpendicular side through origin makes 30° + 90° = 120° with x-axis.</p><p><strong>Step 2:</strong> Let the four vertices be O (origin), A, B, C where:</p><ul><li>O = (0, 0)</li><li>A lies on the side at 30°: A = (2cos30°, 2sin30°) = (√3, 1)</li><li>C lies on the side at 120°: C = (2cos120°, 2sin120°) = (-1, √3)</li><li>B is the fourth vertex (opposite to O): B = A + C = (√3 - 1, 1 + √3)</li></ul><p><strong>Step 3:</strong> All vertices lie above or on x-axis (y ≥ 0). Sum of x-coordinates:</p><p>x-sum = 0 + √3 + (-1) + (√3 - 1) = 2√3 - 2 = 2(√3 - 1)</p><p><strong>Step 4:</strong> Simplifying: 2√3 - 2 ≈ 3.464 - 2 = 1.464</p><p>∴ Answer: C</p>
Correct Answer: C