The smaller side of the rectangle is equal to a, and the acute angle

The smaller side of the rectangle is equal to a, and the acute angle between its diagonals is 60 degrees. Find the diameter of the circle around the rectangle.

The intersection point of the diagonals in the rectangle halves the diagonals. Thus, all four triangles formed at the intersection of the diagonals are isosceles (with equal sides).

The angle at the apex of a triangle with base a is 60 °. The sum of all angles in a triangle is always 180 °. In an isosceles triangle, the angles at the base are equal. Let us find the sum of the angles at the base a:

180 ° – 60 ° = 120 °;

Then each of the angles at the base a is equal to:

120 ° / 2 = 60 °.

All angles in a triangle with base a are 60 °. This means that this triangle is equilateral. This means that all halves of the diagonals of the rectangle are equal to its width and equal to a.

The radius of a circle circumscribed about a rectangle is equal to half of its diagonal:

r = a;

The diameter of a circle is equal to its two radii:

d = 2r;

d = 2a.

Answer: 2a