# The parallelogram has a perimeter of 20 m and an area of 12 m2. Find the sides of the parallelogram

**The parallelogram has a perimeter of 20 m and an area of 12 m2. Find the sides of the parallelogram, considering that the acute angle of the parallelogram is 30 degrees.**

1. Vertices of the parallelogram A, B, C, D. Angle A = 30 degrees.

2. We take the length of the AB side as x, and the length of the AD side as y.

3. From the top B draw the height BH.

4. In triangle ABH, the height BH is the leg opposite an angle of 30 degrees. Therefore, BH = AB / 2 = x / 2 m.

5. Let’s compose two equations:

(1) 2x + 2y = 20; x + y = 10; x = 10 – y;

(2) xy / 2 = 12; xy = 24;

6. Substitute the value x = (10 – y) into the second equation:

(10 – y) y = 24;

10y – y² = 24;

y² – 10y + 24 = 0;

The first value of y = (10 + √100 – 96) / 2 = 6.

The second value is y = (10 – √100 – 96) / 2 = 4.

The first value is x = 10 – 6 = 4.

The second value is x = 10 – 4 = 6.

The length of the AB side can be 4 m or 6 m.

The length of the AD side can be 6 m or 4 m.

Answer: the length of the AB side can be 4 m or 6 m, the length of the AD side can be 6 m or 4 m.