Question

Two persons are X metres apart and the height of one is three times the other.If from the middle point of the line joining their feet an observer finds the angle of elevation of their heads to be complementary then find the height of each person.

Answer 1

Solution:

Let CD=Height of Person A=H meter

AB=Height of person B= 3 H meter

Let E be the point from which angle of elevation is made.

Let, ∠DEC=A, then ∠AEB=90-A

As, both the angles are complementary.

Also, BC=x meter

CE=BE $=\frac{x}{2}$

In right triangle ABE and DCE, right angled at B and C, respectively

$tan(90-A)=\frac{3H}{\frac{x}{2}}\\\\cotA=\frac{6H}{x}\\\\ tan A=\frac{H}{\frac{x}{2}}\\\\tan A=\frac{2H}{x}\\\\ tan A\times cot A=\frac{6H}{x}\times\frac{2H}{x}\\\\ x^2=12 H^2\\\\ H^2=\frac{x^2}{12}\\\\H=\frac{x}{2\sqrt{3}}$

cotA $=\frac{1}{tanA}$, and tan (90-A)=cot A→→Identities  used

Height of two persons are ,  $\frac{x}{2\sqrt{3}} and \frac{3x}{2\sqrt{3}}=\frac{\sqrt{3}x}{2}$ respectively.