Given a rhombus ABCD with sides 12 cm. From the top A to the sides BC and CD two heights are drawn

Given a rhombus ABCD with sides 12 cm. From the top A to the sides BC and CD two heights are drawn, the angle between which is equal to 30 degrees. The perimeter of the rhombus = 48 cm. What is its area?

1. The heights AK and AH are drawn to the sides CD and BC, respectively. The angle between them is ∠HAK = 30 °. S is the area of the rhombus.

2. ∠DAK = 90 ° – ∠HAK = 90 ° – 30 ° = 60 °.

3. We calculate the length of the height of the AK in terms of one of the trigonometric functions ∠DAK (cosine):

AK / AD = cosine ∠DAK = cosine 60 ° = 1/2.

AK = 12 x 1/2 = 6 centimeters.

4. S = CD x AK = 12 x 6 = 72 centimeters².

Answer: S equals 72 centimeters².