One of the sides of the parallelogram is 12, and the other is 5, the angle is 60 °. find S / √3?

We know that one of the sides of the parallelogram is 12, and the other is 5, the angle of 60 ° is also known. There are two ways to solve the problem.

The first solution.

Recall and apply the formula for calculating the area of ​​a parallelogram – this is the product of two sides by the sine of the angle between them.

S = a * b * sin a = 5 * 12 * sin 60 ° = 60 * √3 / 2 = 30√3.

S / √3 = 30√3 / √3 = 30 sq. units.

Second solution.

Let’s draw the height BN.

Consider a triangle:

ABN ^ ∠ N = 90 °, ∠ А = 60 °, therefore ∠ В = 30 °.

АN = AB / 2 = 5/2 as a leg opposite an angle of 30 °.

Let’s apply the Pythagorean theorem to calculate the height:

BN = √ (AB ^ 2 – AN ^ 2) = √ (25 – 25/4) = √ (75/4) = 5√3 / 2.

S = AD * BN = 12 * 5√3 / 2 = 30√3;

S / √3 = 30√3 / √3 = 30 sq. units.