A: If α,β,γ are lengths of altitudes of ΔABC then cosAα+cosBβ+cosCγ=1RR: The altitude of ΔABC through A, B, C are 2Δa,2Δb,2Δc respectively

A : α=2Δa, β=2Δb, γ=2Δc∴ ∑cosAα=∑cosA(2Δa)=∑acosA2Δ=∑2RsinAcosA2Δ=R2Δ∑sin2A=R2Δ(4sinAsinBsinC)=2RsinAsinBsinCΔ=1RR : α=2Δa, β=2Δb, γ=2Δc

A: If α,β,γ are lengths of altitudes of ΔABC then cosAα+cosBβ+cosCγ=1RR: The altitude of ΔABC through A, B, C are 2Δa,2Δb,2Δc respectively

A : α=2Δa, β=2Δb, γ=2Δc∴ ∑cosAα=∑cosA(2Δa)=∑acosA2Δ=∑2RsinAcosA2Δ=R2Δ∑sin2A=R2Δ(4sinAsinBsinC)=2RsinAsinBsinCΔ=1RR : α=2Δa, β=2Δb, γ=2Δc

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A: If $α, β, γ$ are lengths of altitudes of $Δ A B C$ then $\frac{\cos A}{α} + \frac{\cos B}{β} + \frac{\cos C}{γ} = \frac{1}{R}$
R: The altitude of $Δ A B C$ through A, B, C are $\frac{2 Δ}{a}, \frac{2 Δ}{b}, \frac{2 Δ}{c}$ respectively

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A is false, R is true

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Detailed Solution

$A : α = \frac{2 Δ}{a}, β = \frac{2 Δ}{b}, γ = \frac{2 Δ}{c}$
$\begin{array}{l} ∴ \sum \frac{\cos A}{α} = \sum \frac{\cos A}{(\frac{2 Δ}{a})} = \sum \frac{a \cos A}{2 Δ} \\ = \sum \frac{2 R \sin A \cos A}{2 Δ} \\ = \frac{R}{2 Δ} \sum \sin 2 A \\ = \frac{R}{2 Δ} (4 \sin A \sin B \sin C) \\ = \frac{2 R \sin A \sin B \sin C}{Δ} = \frac{1}{R} \end{array}$
$R : α = \frac{2 Δ}{a}, β = \frac{2 Δ}{b}, γ = \frac{2 Δ}{c}$