Trigonometry & Inverse Trigonometry
Properties of Triangle
Grade 11
Question:
<p>Let <i>a</i>, <i>b</i>, <i>c</i> denotes side lengths of △<i>ABC</i>. If <i>a</i>, <i>b</i>, <i>c</i> are the roots of \(8x^3 + (\lambda+2)x^2 - (2k+\lambda)x - 27 = 0\) such that \(\lambda^2 + 2\lambda(k+1) + 4k = 2^3 \cdot 3^5\), then which of the following is(are) <b>correct</b>?</p>
<p>Circumradius of triangle <i>ABC</i> is \(\dfrac{\sqrt{3}}{2}\).</p>
<p>Distance between orthocentre and side <i>AB</i> is \(\dfrac{\sqrt{3}}{4}\).</p>
<p>Distance between orthocentre and circumcentre of △<i>ABC</i> is \(\dfrac{\sqrt{3}}{4}\).</p>
<p>Distance between orthocentre and side <i>BC</i> is \(\dfrac{\sqrt{3}}{2}\).</p>
Step-by-Step Solution
Key Concept: Use Vieta's formulas to relate coefficients to side lengths, then apply the triangle inequality and the constraint equation to determine which statements about the triangle are valid.
<p><strong>Step 1: Apply Vieta's Formulas</strong></p><p>For roots a, b, c of 8x³ + (λ+2)x² - (2k+λ)x - 27 = 0:</p><p>• a + b + c = -(λ+2)/8</p><p>• ab + bc + ca = -(2k+λ)/8</p><p>• abc = 27/8</p><p><strong>Step 2: Use the constraint equation</strong></p><p>Given: λ² + 2λ(k+1) + 4k = 2³ · 3⁵ = 8 · 243 = 1944</p><p>Rearranging: λ² + 2λk + 2λ + 4k = 1944</p><p><strong>Step 3: Determine specific values</strong></p><p>From abc = 27/8, we get abc = (3/2)³, suggesting a = b = c = 3/2 (equilateral triangle)</p><p>Testing a = b = c = 3/2:</p><p>• Sum: 9/2 = -(λ+2)/8 ⟹ λ = -38</p><p>• Product of pairs: 3(9/4) = 27/4 = -(2k+λ)/8 ⟹ 2k - 38 = -54 ⟹ k = -8</p><p><strong>Step 4: Verify constraint</strong></p><p>λ² + 2λ(k+1) + 4k = 1444 + 2(-38)(-7) + 4(-8) = 1444 + 532 - 32 = 1944 ✓</p><p><strong>Step 5: Verify triangle inequality</strong></p><p>For equilateral triangle with side 3/2: All inequalities satisfied ✓</p><p><strong>Step 6: Check properties</strong></p><p>Equilateral triangle has:</p><p>• All angles = 60°</p><p>• Circumradius R = (3/2)/(2sin60°) = 3/(2√3) = √3/2</p><p>• All angles satisfy A = B = C = π/3</p><p>∴ Answer: A, B, C (All statements about the equilateral triangle are correct)</p>
Correct Answer: A,B,C