The angle of the parallelogram is 120 degrees, the large diagonal is 14 cm, and one of the sides

| September 8, 2021 | education

The angle of the parallelogram is 120 degrees, the large diagonal is 14 cm, and one of the sides is 10 cm. Find the perimeter and area of the parallelogram.

Consider a triangle ABC, whose angle B = 120, side AB = 10 cm, and side AC = 14 cm. Using the cosine theorem for triangles, we determine the size of the side BC.

AC ^ 2 = AB ^ 2 + BC ^ 2 – 2 * AB * BC * CosB.

Let the side BC = X cm.

14 ^ 2 = 10 ^ 2 + X ^ 2 – 2 * 10 * X * (-1 / 2).

X ^ 2 + 10 * X – 96 = 0.

Let’s solve the quadratic equation.

D = b ^ 2 – 4 * a * c = 10 ^ 2 – 4 * 1 * (-96) = 100 + 384 = 484.

X1 = (-10 – √484) / (2 * 1) = (-10 – 22) / 2 = -32 / 2 = -16. (Doesn’t fit because <0).

X2 = (-10 _ √484) / (2 * 1) = (-10 + 22) / 2 = 12/2 = 6.

BC = AD = 6 cm.

Determine the perimeter of the parallelogram.

P = AB + BC + CD + AD = 10 + 6 + 10 + 6 = 32 cm.

Determine the area of ​​the parallelogram.

S = AB * BC * SinA = 10 * 6 * √3 / 2 = 30 * √3 cm2.

Answer: The perimeter is 32 cm, the parallelogram area is 30 * √3 cm2.



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