At the rhombus sides are equal and mutually parallel. Its diagonals intersect at right angles and point of intersection are divided into equal parts.
These properties make it easy to find the value of the diagonals of a rhombus.
Let the vertices of a rhombus Latin lettersalphabet A, B, C and D for the convenience of discussion. The point of intersection of the diagonals is traditionally denoted by the letter O. The length of the edges of the rhombus is denoted by the letter a. The value of the angle BCD, which is equal to the angle BAD, denoted α-.
Let us find the value of the short diagonal. Since diagonals intersect at right angles, the COD is a rectangular triangle. Half of the short diagonal OD is the leg of the triangle and can be found through the hypotenuse of CD, and the angle of OCD.
The diagonals of a rhombus are also bisectors of its angles, so OCD angle is α- / 2.
Thus, OD = BD / 2 = CD * sin (α- / 2). That is, the short diagonal BD = 2a * sin (α- / 2).
Similarly, the fact that the COD right-angled triangle, we can express the value of OC (which is half the length of the diagonal).
OC = AC / 2 = CD * cos (α- / 2)
The value of the long diagonal is expressed as follows: AC = 2a * cos (α- / 2)