In the parallelogram ABCD, the sides AB = 4cm, AD = 5√2cm, and the angle A = 45

| September 19, 2021 | education

In the parallelogram ABCD, the sides AB = 4cm, AD = 5√2cm, and the angle A = 45 ° are given. Find the diagonal of the parallelogram and its area.

In a parallelogram, the opposite sides are equal, then AB = CD 4 cm, BC = AD = 5 * √2 cm.

From the triangle ABD, by the cosine theorem, we determine the length of the diagonal BD.

BD ^ 2 = AB ^ 2 + AD ^ 2 – 2 * AB * A * Cos45 = 16 + 50 – 2 * 4 * 5 * √2 * √2 / 2 = 66 – 40 = 26.

ВD = √26 cm.

The sum of the adjacent angles of the parallelogram is 180, then the angle ABC = 180 – 45 = 135.

From the triangle ABC, by the cosine theorem, we determine the length of the diagonal AC.

AC ^ 2 = AB ^ 2 + BC ^ 2 – 2 * AB * BC * Cos135 = 16 + 50 – 2 * 4 * 5 * √2 * (-√2 / 2) = 66 + 40 = 106.

AC = √106 cm.

Determine the area of ​​the parallelogram.

Savsd = AB * AD * Sin45 = 4 * 5 * √2 * √2 / 2 = 20 cm2.

Answer: The area of ​​the parallelogram is 20 cm2, the diagonals are √26 cm, √106 cm.



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