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

June 29, 2021 | education

| **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.

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