Question

Two persons are standing ‘x’ metres apart from each other and the height of the first person is double that of the other. If from the middle point of the line joining their feet an observer finds the angular elevations of their tops to be complementary, then the height of the shorter person (in metres) is 

Options

• `sqrt(2)x`

• `x/(2sqrt(2))`

• `x/sqrt(2)`

• 2x

Answer 1

`x/(2sqrt(2))`

Explanation;

Hint:
Consider the height of the 2^nd person ED be h

Height of the second person is 2h

C is the midpoint of BD

In the right ∆ABC, tan θ = `"AB"/"BC"`

tan θ = `(2"h")/(x/2) = (4"h")/x`  ...(1)

In the right ΔDCE,

tan (90 − θ) =  `"ED"/"CD" = "h"/(x/2)`

cot θ = `(2"h")/x`

⇒ `1/tan theta = (2"h")/x` 

tan θ = `x/(2"h")`  ...(2)

From (1) and (2) we get,

`(4"h")/x = x/(2"h")`

x² = 8h²

⇒ h² = `x^2/8`

h = `sqrt(x^2/8)`

= `x 1/sqrt(4 xx 2)`

= `x/(2sqrt(2))`