<p>In triangle ABC with H as orthocenter at origin (0, 0), B = (-2, 3), and C = (5, -1), find the coordinates of vertex A.</p>
Step-by-Step Solution
Key Concept: In an orthocenter, the line from a vertex to the orthocenter is perpendicular to the opposite side. Use this property with two sides.
<p><strong>Step 1:</strong> Since H is the orthocenter, AH ⊥ BC.</p><p>Slope of BC = \(\frac{-1-3}{5-(-2)} = \frac{-4}{7}\)</p><p><strong>Step 2:</strong> Since AH is perpendicular to BC, slope of AH = \(\frac{7}{4}\)</p><p>If A = (h, k), then \(\frac{k}{h} = \frac{7}{4}\), so \(k = \frac{7h}{4}\)</p><p><strong>Step 3:</strong> Also, CH ⊥ AB. Slope of CH = \(\frac{-1}{5}\)</p><p>Slope of AB = \(\frac{k-3}{h+2}\)</p><p>Since AB ⊥ CH: \(\frac{k-3}{h+2} \cdot \frac{-1}{5} = -1\)</p><p>This gives: \(3\left(\frac{7h}{4} + 1\right) = 2(h-5)\)</p><p><strong>Step 4:</strong> Solving: \(h = -4, k = -7\)</p><p>∴ A = (-4, -7)</p>
<div class="key-concept"><strong>Key Concept:</strong> In an orthocenter, the line from a vertex to the orthocenter is perpendicular to the opposite side. Use this property with two sides.</div>
<div class="trap-box"><strong>Trap:</strong> Students may forget that AH ⊥ BC and CH ⊥ AB simultaneously. Both conditions must be satisfied.</div>
Correct Answer: A(-4, -7)