The parallelogram angle is 120⁰, the sides are 5: 8, and the smaller diagonal is 14cm.

| September 4, 2021 | education

The parallelogram angle is 120⁰, the sides are 5: 8, and the smaller diagonal is 14cm. Find the large diagonal and the area of the parallelogram.

In a parallelogram, the sum of adjacent angles is 180, then the angle BAD = 180 – 120 = 60.

Let the length of the smaller side be 5 * X cm, then the large side is 8 * X cm.

In triangle ABD, by the cosine theorem:

BD ^ 2 = AB ^ 2 + AD ^ 2 – 2 * AB * AD * CosBAD.

196 = 25 * X ^ 2 + 64 * X ^ 2 – 2 * 5 * X * 8 * X * (1/2).

89 * X ^ 2 – 40 * X2 = 196.

49 * X ^ 2 = 196.

X ^ 2 = 4.

X = 2.

Then AB = 2 * 5 = 10 cm, AB = 2 * 8 = 16 cm.

In the triangle ABC, according to the cosine theorem: AC ^ 2 = AB ^ 2 + BC ^ 2 – 2 * AB * BC * Cos120 =

100 + 256 – 2 * 10 * 16 * (-1/2) = 356 + 160 = 516.

AC = √516 = 2 * √129 cm.

Determine the area of ​​the parallelogram. S = AB * AD * Sin60 = 10 * 16 * √3 / 2 = 80 * √3 cm2.

Answer: The large diagonal is 2 * √129 cm, the area is 80 * √3 cm2.



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