Trigonometry & Inverse Trigonometry
Solution of Triangles
Grade 11
Question:
<p>If the circumradius of ∆ABC is 3 units and its area is 6 square units, and ∆DEF is formed by joining the feet of perpendiculars drawn from A, B, C on sides BC, CA, AB respectively, find the perimeter of ∆DEF.</p>
Step-by-Step Solution
Key Concept: The orthic triangle's sides relate to the original triangle through the angles and circumradius. Use the formula for the orthic triangle's perimeter in terms of R and the angles.
<p><strong>Solution:</strong> The triangle DEF formed by the feet of perpendiculars from the vertices is called the orthic triangle. The sides of the orthic triangle are related to the original triangle by: \(d = 2R\sin A\cos A = R\sin 2A\), where R is the circumradius and d is the corresponding side of the orthic triangle.</p><p>Given: \(R = 3\) and Area \(= 6\).</p><p>From Area \(= \frac{abc}{4R}\), we have \(6 = \frac{abc}{12}\), so \(abc = 72\).</p><p>The perimeter of the orthic triangle is \(P_{DEF} = 2R(\sin 2A + \sin 2B + \sin 2C) = 4\).</p>
Correct Answer: 4