Question

In a triangle r is the radius of the incircle, R is the radius of the circumcircle and A is the area of the triangle. If h${}_1$, h${}_2$ and h${}_3$ are the 3 altitudes of the triangle, then $\frac{1}{h_1}+\frac{1}{h_2}+\frac{1}{h_3}=?$

• A. r/A
• B. 1/R
• C. 1/r
• D. R/A

Answer 1

The correct option is C 1/r

Option (c)

Let a = side, h = height,

Since no conditions are mentioned and only variables are used, we can assume an equilateral triangle.

Let r =1. Then R = 2., h = 3. In an equilateral triangle r : R : h = 1 : 2 : 3

Side a = 2 $\sqrt{3}$; Area= $\frac{\sqrt{3}}{4}$ a${}^2$ = 3$\sqrt{3}$.

$\frac{1}{h_1}=\frac{1}{h_2}=\frac{1}{h_3}=\frac{1}{3}$. Sum = 1

Only option c gives 1.