Straight Lines
Perpendicularity Condition
Grade 11
Question:
<p>In a rectangle with vertices at (0, 0), (a, 0), (a, b), and (0, b), if AD ⊥ BE where D and E are midpoints, find the relationship between a and b.</p>
Step-by-Step Solution
Key Concept: For two lines to be perpendicular, the product of their slopes must equal -1. Use the perpendicularity condition on the given lines.
<p><strong>Step 1:</strong> Rectangle ABCD with A(0, b), B(a, 0), C(a, 0), D(0, 0)</p><p><strong>Step 2:</strong> Midpoint D = (a/2, 0), Midpoint E = (a, b/2)</p><p><strong>Step 3:</strong> Slope of AD = \(\frac{b - 0}{0 - a/2} = -\frac{2b}{a}\)</p><p><strong>Step 4:</strong> Slope of BE = \(\frac{b/2 - 0}{a - a} \text{ [undefined, vertical line]}\)</p><p><strong>Step 5:</strong> For AD ⊥ BE: \(\left(-\frac{2b}{a}\right) \cdot \text{(slope of BE)} = -1\)</p><p>\(\left(\frac{b}{a}\right) \cdot \left(\frac{a}{2}\right) = -1\)</p><p>\(\frac{b^2}{a^2} = \frac{1}{2}\)</p><p>\(a = \pm\sqrt{2}b\)</p>
Correct Answer: a = ±√2 b