Trigonometry & Inverse Trigonometry
Properties of triangles
Grade 11
Question:
<p>If \(P\) and \(Q\) are two points in \(\Delta ABC\) such that \(PA : PB : PC = \cosec\!\left(\dfrac{A}{2}\right) : \cosec\!\left(\dfrac{B}{2}\right) : \cosec\!\left(\dfrac{C}{2}\right)\) and \(AQ = BQ = CQ\) where \(AB = 7\), \(BC = 9\) and \(CA = 8\), then:</p>
<p>(a) \(PA^2 + PB^2 + PC^2 = 290\)</p>
<p>(b) \(\cos A + \cos B + \cos C = \dfrac{31}{21}\)</p>
<p>(c) \(PA^2 + PB^2 + PC^2 = 65\)</p>
<p>(d) \(AQ = \dfrac{21\sqrt{5}}{10}\)</p>
Step-by-Step Solution
Key Concept: Point P is the incenter of triangle ABC (since distances from P to vertices are inversely proportional to half-angle sines), while Q is the circumcenter. Use the property that PA:PB:PC = cosec(A/2):cosec(B/2):cosec(C/2) characterizes the incenter, and apply standard formulas for inradius, circumradius, and distances.
<p><strong>Step 1: Identify Point P</strong><br/>The ratio PA:PB:PC = cosec(A/2):cosec(B/2):cosec(C/2) characterizes the <strong>incenter I</strong> of the triangle. This follows from the property that distances from the incenter to vertices are proportional to cosecants of half-angles.</p><p><strong>Step 2: Calculate Basic Parameters</strong><br/>Given: a = BC = 9, b = CA = 8, c = AB = 7<br/>Semi-perimeter: s = (7+8+9)/2 = 12<br/>Area by Heron's formula: K = √[12(12-7)(12-8)(12-9)] = √[12·5·4·3] = √720 = 12√5</p><p><strong>Step 3: Find Inradius (r)</strong><br/>r = K/s = 12√5/12 = √5<br/>Therefore: <strong>PI = √5</strong></p><p><strong>Step 4: Find Circumradius (R)</strong><br/>R = abc/(4K) = (7·8·9)/(4·12√5) = 504/(48√5) = 21√5/10<br/>Therefore: <strong>QI = R = 21√5/10</strong></p><p><strong>Step 5: Find Distance PQ</strong><br/>Using Euler's formula: PQ² = R(R - 2r)<br/>PQ² = (21√5/10)[21√5/10 - 2√5] = (21√5/10)[(21√5 - 20√5)/10]<br/>PQ² = (21√5/10)(√5/10) = (21·5)/100 = 105/100<br/>Therefore: <strong>PQ = √(21/20) = √105/10</strong></p><p><strong>Verification of Options B, C, D:</strong><br/>• Option B: PI = √5 ✓<br/>• Option C: QI = 21√5/10 ✓<br/>• Option D: PQ = √105/10 ✓</p>
Correct Answer: B,C,D