Adjacent sides of a parallelogram are equal and one of diagonals is equal to any one side of the parallelogram.Find the ratio of its diagonals )

Question

It's hard

Answer 1

If adjacent sides of a parallelogram are equal, then the parallelogram is a rhombus.  In a rhombus, the diagonals bisect each other and are perpendicular to each other.  

 

Let the lengths of the diagonals be x and y, with x being the length of the diagonal which is the same as the lengths of the two equal sides.

 

Then, the rhombus can be divided into 4 congruent right triangles with hypotenuse x and legs x/2 and y/2.

 

By the Pythagorean Theorem, (x/2)² + (y/2)² = x².

 

So, (3/4)x² = (1/4)y²

 

         √3x = y

 

y/x = √3   (or equivalently, x/y = √3/3)