Straight Lines
Coordinate Geometry of Polygons
Grade 11
Question:
<p>Let the opposite angular points of a square be <span class="math">(3, 4)</span> and <span class="math">(1, -1)</span>. Then, the coordinates of the remaining angular points are</p>
<p>(a) <span class="math">\left(\frac{9}{2}, \frac{1}{2}\right)</span> and <span class="math">\left(-\frac{1}{2}, \frac{5}{2}\right)</span></p>
<p>(b) <span class="math">\left(\frac{9}{2}, -\frac{1}{2}\right)</span> and <span class="math">\left(-\frac{1}{2}, \frac{5}{2}\right)</span></p>
<p>(c) <span class="math">\left(-\frac{9}{2}, \frac{1}{2}\right)</span> and <span class="math">\left(-\frac{1}{2}, \frac{5}{2}\right)</span></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: In a square, the diagonals bisect each other at right angles and have equal length. If two opposite vertices are given, their midpoint is the center, and the other two vertices can be found using the perpendicularity and equal length properties of diagonals.
<p><strong>Step 1: Find the center of the square.</strong></p><p>Let A = (3, 4) and C = (1, -1) be opposite vertices. The center O is the midpoint:</p><p>O = ((3+1)/2, (4-1)/2) = (2, 3/2)</p><p><strong>Step 2: Find the slope of diagonal AC.</strong></p><p>Slope of AC = (-1-4)/(1-3) = -5/(-2) = 5/2</p><p><strong>Step 3: Find the slope of diagonal BD.</strong></p><p>Since diagonals of a square are perpendicular, if slope of AC = 5/2, then slope of BD = -2/5</p><p><strong>Step 4: Use symmetry property.</strong></p><p>The diagonals are equal in length. The distance from A to O equals the distance from B to O (and similarly for D).</p><p>Distance AO = √[(3-2)² + (4-3/2)²] = √[1 + (5/2)²] = √[1 + 25/4] = √(29/4) = √29/2</p><p><strong>Step 5: Find the other two vertices B and D.</strong></p><p>The unit vector along BD direction with slope -2/5: direction vector is (5, -2), so unit vector = (5/√29, -2/√29)</p><p>B = O + (√29/2) × (5/√29, -2/√29) = (2, 3/2) + (5/2, -1) = (9/2, 1/2)</p><p>D = O - (√29/2) × (5/√29, -2/√29) = (2, 3/2) - (5/2, -1) = (-1/2, 5/2)</p><p><strong>Step 6: Verify the solution.</strong></p><p>Check that B and D are equidistant from O and that the four points form a square with perpendicular diagonals. ✓</p><p><strong>∴ Answer: a</strong></p>
Correct Answer: a