Question

length and breadth if rectangle paper is in the ratio is √3:1.then what is the angle made by its diagonals​

Answer 1

Given :

The ratio of length and breadth of rectangle = L : B = √3 : 1

To Find :

The angle made by diagonals

Solution :

Let The Length = L = √3 x

     The breadth = B = x

From figure ABCD is rectangle

For right angle triangle Each angle of rectangle = 90°

In Δ  ABC

Sin angle = $\dfrac{perpendicular}{hypotenuse}$

Or, Sin 90° = $\dfrac{AB}{AC}$

Or, 1 = $\dfrac{x}{AC}$

Or, AC = x  

Again

∵  Tan angle  = $\dfrac{perpendicular}{base}$

$Tan\Theta _2$ = $\dfrac{AB}{BC}$

          = $\dfrac{x}{\sqrt{3} x}$                    

          = $\dfrac{1}{\sqrt{3} }$

∴  $\Theta _2$ = $Tan^{-1} ( \dfrac{1}{\sqrt{3} } )$

       =  30°

For any triangle , sum of all angles = 180°

So,  $\Theta _1$  = 180° - ( 90° +   $\Theta _2$ )

            = 180° - ( 90° +   30° )

            = 60°

Hence, The angle of diagonal are 30° , 60°    Answer