Question

If P₁, P₂, P₃ are respectively the perpendiculars from the vertices of a triangle to the opposite sides, then `cosA/P_1 + cosB/P_2 + cosC/P_3` is equal to

विकल्प

• `1/r`

• `1/R`

• `R/Δ`

• None of these

Answer 1

`1/R`

Explanation:

We have, Δ = `1/2 ap_1 = 1/2 bp_2 = 1/2 cp_3`

∴ `1/P_1 = a/(2Δ), 1/P_2 = b/(2Δ), 1/P_3 = c/(2Δ)`

`cosA/P_1 + cosB/P_2 + cosC/P_3 = 1/(2Δ)` ......(a cos A + b cos B + c cos C)

= `R/Δ` (sin A cos A + sin B cos B + sin C cos C)

= `R/(2Δ)` (sin 2A + sin 2B + sin 2C)

= `R (4 sin A sin B sin C)/(2Δ)`

= `(2R)/(((abc)/(4R))) xx a/(2R) xx b/(2R) xx c/(2R)`

= `1/R`