+92 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 $R^2.$
+421 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 $3.$
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
+128408 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
