The diagonal AC of rectangle ABCD is 3 cm and makes an angle of 60 with side AD. Find the area of rectangle ABCD.

Rectangle – a polygon in which all four corners are straight and opposite sides are equal:

AB = CD;

BC = AD.

In order to find out the area of ​​a rectangle, you need to multiply its length by width:

S = a b.

In order to find the length and width of a given rectangle, consider the triangle ΔABC.

Knowing the value of the angle ∠A and the hypotenuse AC, we apply the cosine theorem, where the cosine is the ratio of the adjacent leg to the hypotenuse:

sin A = AD / AC;

AD = AC sin A;

sin 60º = 1/2;

AD = 3 1/2 = 3/2 = 1.5 cm.

To calculate the length of the leg CD, we use the Pythagorean theorem:

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

CD ^ 2 = AC ^ 2 – AD ^ 2;

CD ^ 2 = 32 – 1.52 = 9 – 2.25 = 6.75;

СD = √6.75 ≈ 2.6 cm.

S = 1.5 2.6 = 3.9 cm2.

Answer: the area of ​​the rectangle is 3.9 cm2.