# The diagonals of the parallelogram are 4cm and 4√2cm, and the angle between them is 45 degrees.

The diagonals of the parallelogram are 4cm and 4√2cm, and the angle between them is 45 degrees. Find the area and perimeter.

Determine the area of the parallelogram.

Savsd = АС * ВD * Sin45 = 4 * √2 * 4 * √2 / 2 = 16 cm2.

By the cosine theorem, we determine the length of the side AB.

AB ^ 2 = OA ^ 2 + BO ^ 2 – 2 * AO * BO * Cos45 = (2 * √2) ^ 2 + 2 ^ 2 – 2 * √2 * 2 * √2 / 2 = 8 + 4 – 4 = 8.

AB = CD = √8 = 2 * √2.

Let us determine the length of the BC side.

ВС ^ 2 = OC ^ 2 + ОВС ^ 2 – 2 * OC * ОВ * Cos135 = (2 * √2) ^ 2 + 2 ^ 2 – 2 * √2 * 2 * (-√2 / 2) = 8 + 4 + 4 = 16.

BC = 4 cm.

Let’s define the perimeter. P = 2 * (AB + BC) = 2 * (2 * √2 + 4) = 8 + 4 * √2 cm.

Answer: The area is 16 cm2, the perimeter is 8 + 4 * √2 cm.

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