Find the diagonals of a rectangle, ABCD if CAD = 30 degrees, A side CD = 4cm.

The diagonals of the rectangle ABCD divide it into two equal right-angled triangles ABC and ACD. The diagonal ac is the hypotenuse of any of these triangles. Let’s consider one of them, let it be ΔACD.

The length of the hypotenuse of a right triangle can be calculated through the sine of an acute angle and the length of the opposite leg. Thus, if:

sin A = CD / AC, then

AC = CD / sin A, where

CD = 4 cm;

∠CAD = 30 °;

sin 30 ° = 1/2;

AC = 4: 1/2 = 4/1: 1/2 = 4/1 • 2/1 = 8/1 = 8.

Since the diagonals of the rectangle are equal, then AC = BD = 8 cm.

Answer: the length of the diagonals of the rectangle is 8 cm.