The perimeter of the rhombus is 20 cm and one of its diagonals is 8 cm, find the second diagonal.

A rhombus is a parallelogram in which all sides are equal and the angles are not right.

Since the perimeter of the rhombus is 68, and all its four sides are equal, then:

AB = BC = CD = AD = P / 4;

AB = BC = CD = AD = 20/4 = 5 cm.

The diagonals of the rhombus are perpendicular and the intersection point is halved:

AO = OC = AC / 2;

AO = OC = 8/2 = 4 cm;

BO = OD = BD / 2.

In order to find the length of the diagonal BD, consider the triangle ΔABO. This triangle is rectangular. We apply the Pythagorean theorem, according to which the square of the hypotenuse is equal to the sum of the squares of the legs:

AB ^ 2 = BO ^ 2 + AO ^ 2;

BO ^ 2 = AB ^ 2 – AO ^ 2;

BО ^ 2 = 5 ^ 2 – 4 ^ 2 = 25 – 16 = 9;

BО = √9 = 3 cm;

BD = BO + OD;

ВD = 3 + 3 = 6 cm.

Answer: the length of the diagonal BD is 6 cm.