Question

$\text{In the adjacent figure, 'P' is any arbitrary interior of the triangle ABC,}$ H_a, H_b, H_c $\text{are the length of altitudes drawn from vertices A, B and C respectively. If}$ x_a, x_b and x_c $\text{represent the distance of 'P' from sides BC, AC and AB respectively, then}$

$\frac{x_a}{H_a}+\frac{x_b}{H_b}+\frac{x_c}{H_c}\text{is always equal to}$

• A. 3
• B. 2
• C. 1
• D. 1/3

Answer 1

The correct option is C

1

$\text{We have},\Delta ={\Delta }_{BPC}+{\Delta }_{APC}+{\Delta }_{APB}\Rightarrow =\Delta =\frac{1}{2}a{x}_a+\frac{1}{2}b{x}_b+\frac{1}{2}c{x}_{c}Also,\Delta =\frac{1}{2}a{H}_a=\frac{1}{2}b{H}_b=\frac{1}{2}c{H}_c\Rightarrow \Delta =\Delta (\frac{x_a}{H_a}+\frac{x_b}{H_b}+\frac{x_c}{H_c})\Rightarrow =\frac{x_a}{H_a}+\frac{x_b}{H_b}+\frac{x_c}{H_c}=1$