Question

The product of the arithmetic mean of the lengths of the sides of a triangle and harmonic mean of the lengths of the altitudes of the triangle is equal to: 

[where ∆ is the area of the triangle ABC] 

(A) ∆ 

(B) 2∆ 

(C) 3∆ 

(D) 4∆

Answer 1

Correct option (B) 2∆

Explanation:

Arithmetic mean = (a + b + c)/3

= 2R/3(sinA + sinB + sinC)

Length of altitude = b sinC = 2R sinBsinC

Harmonic mean

Product = (6R)(2R/3)(sinAsinBsinC)

 = 4R²sinA sinBsinC

= (a/sinA)(b/sinB)sinAsinBsinC = 2∆