In rectangle ABCD, AC = 14 (diagonal), angle ACD = 60 degrees, find the area of the rectangle.

A rectangle is a quadrangle in which opposite sides are equal, and all corners are right:

AB = CD;

BC = AD.

The area of ​​a rectangle is the product of its length and width:

S = a * b, where:

S is the area of ​​the rectangle;

a – the length of the rectangle;

b is its width.

In order to calculate the length and width of the rectangle, consider the triangle ΔАСD.

To calculate the CD side, we apply the cosine theorem:

cos C = CD / AC;

CD = AC · cos C;

cos 60º = 1/2 = 0.5;

СD = 14 0.5 = 7 cm.

To calculate the side AD, we use the Pythagorean theorem:

AC ^ 2 = CD ^ 2 + AD ^ 2;

AD ^ 2 = AC ^ 2 – CD ^ 2;

AD ^ 2 = 142 – 72 = 196 – 49 = 147;

AD = √147 ≈ 12.1 cm.

S = 7 12.1 = 84.7 cm2.

Answer: the area of ​​the rectangle is 84.7 cm2.