In a rectangle, one of the angles formed by the diagonals is 120 (degrees), the diagonals

In a rectangle, one of the angles formed by the diagonals is 120 (degrees), the diagonals of the rectangle are 18 cm. How many centimeters is the smallest side of the rectangle?

1. AC and BD are the diagonals of the rectangle ABCD. O is the point of their intersection. ∠AOD = 120 °.

2. АВ = 180 ° – AOD = 180 ° – 120 ° = 60 °.

3.In ΔAOB, the sides BO and AO are equal, since the diagonals at point O are divided in half. The angles at the side AB are also equal:

∠ABO = ∠BAO = (180 ° – 60 °) / 2 = 60 °.

4. ΔAOB – equilateral, since all its angles are equal to 60 °. Hence, all its sides are equal:

AB = BO = AO = 18: 2 = 9 cm.

5. АD = √ВD² – AB² = √18² – 9² √324 – 81 = √243 = 9√3 cm.

Answer: AB = 9 cm – the smaller side of the rectangle.