# 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.

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