Question

If two towers of height h1 and h2 subtend angle 60° and 30° respectively at the mid ponit of line joining their bases then h1:h2 is?

Answer 1

Question:

If two towers of height h1 and h2 subtend angle 60° and 30° respectively at the mid point of line joining their bases then find the ratio of their heights, ie; h1:h2 .

Answer:

h1/h2 = 3:1

Note:

• tan@ = perpendicular/base

• tan0° = 0

• tan30° = 1/√3

• tan45° = 1

• tan60° = √3

• tan90° = ∞

Solution:

Let's plot a rough diagram to describe the given situation.

Let AB and DC be two towers with A and D as their foots respectively.

Also, let the towers AB and DC have the heights h1 and h2 respectively.

Let, point O be the mid point of the line joining from A to D , such that OA = OD = x .

{ For diagram, please refer to the attached }

Now,

In ∆AOB ,

=> tan60° = AB/OA

=> √3 = h1/x

=> h1 = √3x --------(1)

Now,

In ∆DOC ,

=> tan30° = CD/OD

=> 1/√3 = h2/x

=> h2 = x/√3 ---------(2)

Now,

Dividing eq-(1) by eq-(2) , we get ;

=> h1/h2 = (√3x)/(x/√3)

=> h1/h2 = √3•√3•x/x

=> h1/h2 = 3

=> h1/h2 = 3/1

=> h1:h2 = 3:1

Hence,

The required ratio of their heights is ;

h1:h2 = 3:1

Answer 2

$\huge{\orange{\underline{\red{\mathbb{ANSWER:-}}}}}$

H1:H2=3:1.

$\large{\orange{\underline{\red{\mathbb{STEP \: BY \:STEP \:EXPLANATION:- }}}}}$

Let AB be the tower of height h1 m and CD be the tower of height h2 m.

Let P be the midpoint of the line BC. Then ∠APB = 60° & ∠DPC= 30°

Let BP = PC = x

In ∆ APB ,

$\huge\red\rightarrow$tan 60°  = AB / BP ( perpendicular/ base)

$\huge\red\rightarrow$tan 60 = h1 / x

$\huge\red\rightarrow$√3 = h1 / x

$\huge\red\rightarrow$h1= √3 x…………….( 1 )

In ∆DPC

$\huge\red\rightarrow$tan 30° = DC/PC

$\huge\red\rightarrow$1/√3 = h2 / x

$\huge\red\rightarrow$h2= x/√3…………( 2 )

From eq  ( 1 ) & ( 2 )

$\huge\red\rightarrow$h1/h2=√3x /(x/√3)

$\huge\red\rightarrow$h1/h2= (√3 )(√3 ) / 1

$\huge\red\rightarrow$3 /1 = h1 / h2

$\huge\red\rightarrow$h1 : h2 = 3:1

Hence, the required ratio OF h1:h2=3:1.

$\huge{\orange{\underline{\red{\mathbb{THANKS.}}}}}$