# In a parallelogram, the side and the large diagonal are 3 and √37, respectively, find the perimeter

**In a parallelogram, the side and the large diagonal are 3 and √37, respectively, find the perimeter of the parallelogram if its acute angle is 60 degrees**

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

In the triangle ABC, we apply the cosine theorem and determine the length of the BC side.

AC ^ 2 = AB ^ 2 + BC ^ 2 – 2 * AB * BC * Cos120.

37 = 9 + BC ^ 2 – 2 * 3 * BC * (-1/2).

BC^2 + 3 * BC – 28 = 0.

Let’s solve the quadratic equation.

D = b ^ 2 – 4 * a * c = 3 ^ 2 – 4 * 1 * (-28) = 9 + 112 = 121.

BC1 = (-3 – √121) / (2 * 1) = (-3 – 11) / 2 = -14 / 2 = -7.

BC2 = (-3 + √121) / (2 * 1) = (-3 + 11) / 2 = 8/2 = 4.

Then Ravsd = 2 * (AB + BC) = 2 * (3 + 4) = 14 cm.

Answer: The perimeter of the parallelogram is 14 cm.