The parallelogram angle is 60 °, the side difference is 4 cm, and the large diagonal is 14 cm

| August 13, 2021 | education

The parallelogram angle is 60 °, the side difference is 4 cm, and the large diagonal is 14 cm Find the smaller diagonal and the area of the parallelogram

Let’s designate the side AB of the parallelogram through X m, then, by condition, the side AD = (X + 4).

Since the sum of the adjacent angles of the parallelogram is 180, the angle ADC = 180 – 60 = 120.

We will use the cosine theorem for the triangle ACD.

AC ^ 2 = X ^ 2 + (X + 4) ^ 2 – 2 * X * (X + 4) * Cos120.

196 = X ^ 2 + X ^ 2 + 8 * X + 16 – 2 * X * (X + 4) * (-1 / 2).

196 = 2 * X ^ 2 + 8 * X + 16 + X ^ 2 + 4 * X.

3 * X ^ 2 + 12 * X – 180 = 0.

X ^ 2 + 4 * X – 60 = 0.

Let’s solve the quadratic equation.

D = b ^ 2 – 4 * a * c = 42 – 4 * 1 * (-60) = 16 + 240 = 256.

X1 = (-4 – √256) / (2 * 1) = (-4 – 16) / 2 = -20 / 2 = -10. (Doesn’t fit because <0).

X1 = (-4 + √256) / (2 * 1) = (-4 + 16) / 2 = 12/2 = 6.

AB = 6 cm, then AD = 6 + 4 = 10 cm.

From the triangle ABD, by the cosine theorem, we define the smaller diagonal BD.

BD ^ 2 = AB ^ 2 + AD ^ 2 – 2 * AB * AD * Cos60 = 36 + 100 – 2 * 6 * 10 * 1/2 = 136 – 60 = 76.

ВD = √76 = 2 * √19 cm.

We will draw the height of the ВН and from the right-angled triangle ABН we will determine the leg of the ВН.

ВН = AB * Sin60 = 6 * √3 / 2 = 3 * √3 cm.

Determine the area of ​​the parallelogram.

S = AD * BH = 10 * 3 * √3 = 30 * √3 cm2.

Answer: The smaller diagonal is 2 * √19 cm, the area is 30 * √3 cm2.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.