Vector Algebra
Lami's Theorem
Grade 12
Question:
<p>Three forces <em>P</em>, <em>Q</em>, <em>R</em> act along the bisectors of the angles of a triangle <em>ABC</em>. By Lami's theorem, which of the following is correct?</p>
<p>\(\dfrac{P}{\sin A/2} = \dfrac{Q}{\sin B/2} = \dfrac{R}{\sin C/2}\)</p>
<p>\(\dfrac{P}{\cos A/2} = \dfrac{Q}{\cos B/2} = \dfrac{R}{\cos C/2}\)</p>
<p>\(\dfrac{P}{\tan A/2} = \dfrac{Q}{\tan B/2} = \dfrac{R}{\tan C/2}\)</p>
<p>\(\dfrac{P}{\cos A} = \dfrac{Q}{\cos B} = \dfrac{R}{\cos C}\)</p>
Step-by-Step Solution
Key Concept: Lami's theorem applies when three forces in equilibrium act along lines through a common point; here the angle bisectors of a triangle meet at the incenter, and the forces must be proportional to sines of angles between their lines of action (which are the angles of the triangle).
Step 1: Identify that three forces P, Q, R act along the angle bisectors of triangle ABC, which are concurrent at the incenter I. Step 2: Apply Lami's theorem: When three coplanar forces acting at a point are in equilibrium, each force is proportional to the sine of the angle between the other two forces. Step 3: The angle between the bisector of angle A and bisector of angle B is (π - C)/2 + (π - C)/2 = π - C, but more directly: the angles between consecutive bisectors at the incenter are π/2 + A/2, π/2 + B/2, and π/2 + C/2 for opposite angles. Step 4: By Lami's theorem: P/sin(π/2 + A/2) = Q/sin(π/2 + B/2) = R/sin(π/2 + C/2) Step 5: Since sin(π/2 + θ) = cos(θ), we get: P/cos(A/2) = Q/cos(B/2) = R/cos(C/2) ∴ Answer: B
Correct Answer: B