Find the area of the rectangle if one of the sides is 5cm and the angle between the diagonals is 60 degrees.

1. Vertices of the rectangle A, B, C, D. AB = 5 cm. O – the point of intersection of the diagonals AC and BD.

∠АВ = 60 °.

2. Triangle AOB – isosceles, since the diagonals AC and BD are divided by the point of intersection. That is, BO = AO and ∠ABO = ∠BAO.

3. ∠ABO = ∠BAO = (180 ° – 60 °) / 2 = 60 °. All angles of the ABO triangle are equal. Therefore, the indicated triangle is equilateral. That is, AB = BO = 5 cm.

4. ВD = BО x 2 = 5 x 2 = 10 cm.

5. We calculate the length of the side AD of the rectangle, which in the right-angled triangle ABD is the leg:

АD = √ВD² – AB² = √10² – 5² = √100 – 25 = 5√3 cm.

6. Calculate the area (S) of the rectangle:

S = AB x AD = 5 x 5√3 = 25√3 cm².

Answer: the area of ​​the rectangle is 25√3 cm².