The parallelogram angle is 120 degrees, the sides are 5: 8, and the smaller diagonal is 14 cm.

| March 17, 2021 | education

The parallelogram angle is 120 degrees, the sides are 5: 8, and the smaller diagonal is 14 cm. Find the large diagonal and the area of the parallelogram.

Let the smaller side be 5 * X cm, then the larger side is 8 * X.

Consider a triangle ABD, whose angle A = 180 – 120 = 60, side AB = 5 * X cm, and side AD = 8 * X cm. Using the cosine theorem for triangles.

BD ^ 2 = AB ^ 2 + BD ^ 2 – 2 * AB * BD * CosA.

142 = (5 * X) ^ 2 + (8 * X) ^ 2 – 2 * (5 * X) * (8 * X) * (1/2).

89 * X ^ 2 – 40 * X ^ 2 = 196.

49 * X ^ 2 = 196.

X ^ 2 = 196/49 = 4.

X = 2 cm.

Then AB = CD = 5 * 2 = 10 cm.

AD = BC = 8 * 2 = 16 cm.

Determine the area of ​​the parallelogram.

S = AB * AD * SinA = 10 * 16 * √3 / 2 = 80 * √3 cm2.

Since the sum of the squares of the diagonals of a parallelogram is equal to twice the sum of the squares of its two adjacent sides (AC² + BD²) = 2 * (AB² + AD²), then

AC ^ 2 = 2 * (100 + 256) – 196 = 516.

AC = √516 = 2 * √129 cm.

Answer: AC = 2 * √129 cm, the parallelogram area is 80 * √3 cm2.



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