The perimeter of the parallelogram is 60 cm. Find the lengths of the sides if you know that the diagonal

The perimeter of the parallelogram is 60 cm. Find the lengths of the sides if you know that the diagonal of the parallelogram divides the angle into parts 30 ° and 90 °.

Let us introduce the notation: ABCD – parallelogram, BD – its diagonal, B – obtuse angle of the parallelogram, A – acute angle.

If the diagonal BD divides the angle B into parts equal to 30 ° and 90 °, then ∠B = 90 ° + 30 ° = 120 °. Since the sum of two adjacent angles of a parallelogram is 180 °, then ∠A = 180 ° – ∠B = 60 °.

Obviously, the diagonal BD is perpendicular to the side AB, hence the triangle ABD is rectangular with hypotenuse AD and legs AB and BD.

The opposite sides of the parallelogram are equal to each other, therefore the sum of two adjacent sides is equal to half of the perimeter: AB + AD = 60/2 = 30 cm, AD = 30 – AB.

AB – leg adjacent to angle A, the ratio of the adjacent leg to the hypotenuse is equal to the cosine of the angle.

AB / AD = cos A;

AB / (30 – AB) = cos 60 ° = 0.5.

2 * AB = 30 – AB;

3 * AB = 30;

AB = 30/3 = 10 cm;

AD = 30 – 10 = 20 cm.

The sought lengths of the sides of this parallelogram are 10 cm and 20 cm.