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The diagonal of a rectangle and the diagonal of a square have the same length. The diagonal of the rectangle forms a $60^\circ$ angle with one of its sides. $R$ is the ratio of the area of the rectangle to that of the square. Find $R^2.$

 
 Nov 21, 2020
 #1
avatar+260 
-2

Let the diagonal of the square be $x.$

 

Then we have the long side of the rectangle = $x/2.$

 

We have the short side of the rectange is $x/ \sqrt{3}.$

 

We also have the side of the square is $x\sqrt{2}/2.$

 

So the area of the square is $x^2/2$ and the area of the rectangle is $\frac{x^2}{2\sqrt{3}}.$ 

 

So, the ratio^2 is $3.$

 
 Nov 21, 2020
 #2
avatar
+2

Let the length of diagonal be 10

 

Square area is 102/2 = 50

 

The diagonal and 2 sides of a rectangle form a 30-60-90 triangle.

 

The lengths of  both legs are         sin30º * 10 = 5             sin60º * 10 = 8.660254038

 

The area of the rectangle is             5 * 8.660254038 = 43.30127019

 

R2 = (43.30127019 / 50)2     ==>  R2 = 0.75    or   3/4

 
 Nov 21, 2020
 #3
avatar+112375 
+1

Let the diagonal  length of  both  =  D

 

In the rectangle...  the side  opp the 30° angle  = D/2 and .the side  opp the 60° angle  =    D√3/2   and

 

The area of the rectangle  =  D^2 [√3/4]

 

In the square......the side length =  D/√2

 

Area of square  = (D / √2)^2  = D^2/2

 

R =  D^2 [ √3/4]  / [D^2 / 2 ]   =  (√3/2)

 

R^2 =  3/4  

 

EDIT TO CORRECT A PRIOR MISTAKE

 

 

cool cool cool

 
 Nov 22, 2020
edited by CPhill  Nov 22, 2020
edited by CPhill  Nov 22, 2020
 #4
avatar+452 
+1

let diagonal  be            d = 2

 

square, area                 As = 2 u2

 

rectangle, area             Ar = √3

 

ratio               R = Ar / As = √3 / 2 = 0.866025403

 

R2 = 0.75  ==> 3/4 

 
 Nov 22, 2020

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