Trigonometry & Inverse Trigonometry
Triangle geometry
Grade 11
Question:
<p>In a triangle ABC, medians AD and BE are drawn. If $AD = 4$; $\angle DAB = \frac{\pi}{6}$ and $\angle ABE = \frac{\pi}{3}$ then the area of the triangle ABC is:</p>
<p>(a) $\frac{8}{3\sqrt{3}}$</p>
<p>(b) $\frac{16}{3\sqrt{3}}$</p>
<p>(c) $\frac{32}{3\sqrt{3}}$</p>
<p>(d) $\frac{64}{3\sqrt{3}}$</p>
Step-by-Step Solution
Key Concept: Use properties of medians and the given angle conditions to set up area relationships.
<p>Using the property that medians divide a triangle into six smaller triangles of equal area, and applying the given angle conditions with median length AD = 4.</p><p>Through geometric relations and the angle constraints, the area works out to $\frac{16}{3\sqrt{3}} = \frac{16\sqrt{3}}{9}$.</p>
Correct Answer: B